the schur functors and the resolution of determinantal...

Treball final de grau

GRAU DE MATEMÀTIQUES

Facultat de Matemàtiques i InformàticaUniversitat de Barcelona

The Schur functors and theresolution of determinantal

varieties.

Autor: Liena Colarte Gómez.

Director: Dra. Rosa Maria Miró-Roig.Realitzat a: Departament de Matemàtiques i Informàtica.

Barcelona, January 17, 2017

Contents

Introduction iii

1 Preliminaries. 11.1 Multilinear algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Hopf algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 The Exterior Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 The Symmetric Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 The Divided Power Algebra. . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Combinatorics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Partitions and skew partitions. . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Tableaux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Schur Functors and CoSchur Functors. 92.1 Schur Functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 The freeness of Schur Functors. . . . . . . . . . . . . . . . . . . . . . . 112.2 CoSchur Functors and the freeness of the CoSchur functors. . . . . . . . . . . 192.3 Decomposition of Schur Functors. . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Cauchy Decomposition Formulas for Schur Functors. . . . . . . . . . . . . . . 25

2.4.1 The decomposition of the Symmetric Algebra. . . . . . . . . . . . . . . 252.4.2 The decomposition of the exterior algebra. . . . . . . . . . . . . . . . . 27

2.5 The Littlewood-Richardson rule for Schur functors. . . . . . . . . . . . . . . . 282.5.1 The Schensted Process and Words of Yamanouchi. . . . . . . . . . . . 282.5.2 The Littlewood-Richardson Rule. . . . . . . . . . . . . . . . . . . . . . . 35

3 A minimal free complex associated to the minors of a matrix. 413.1 A minimal free complex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 The group algebra and combinatorics. . . . . . . . . . . . . . . . . . . . 423.1.2 The complex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1.3 The Proof of Lemma 3.1.31. . . . . . . . . . . . . . . . . . . . . . . . . . 493.1.4 The proof of Lemma 3.1.32. . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Determinantal ideals and determinantal varieties. . . . . . . . . . . . . . . . . 543.2.1 A minimal free resolution of determinantal ideals. . . . . . . . . . . . 56

Bibliography 67

i

Introduction.

Resolutions is one of the most effective methods to obtain information about varietiesin Algebraic Geometry. For many years there has been considerable efforts in finding aresolution of determinantal varieties. To put the problem plainly, assume R = K[x0, . . . , xs]is the polynomial ring over an algebraically closed field of characteristic zero and Ps is theprojective space of dimension s over K. Given (ri,j) a homogeneous matrix of size p× qwith entries in R, the problem is to find an explicit minimal free resolution of the ideal Itdefined by the t× t minors of this matrix. Over certain hypothesis on It, this is a minimalfree resolution of the variety X = {z ∈ Ps | rg((ri,j)(z)) < t} of Ps. It provides the Hilbertpolynomial of X, the projective dimension and the arithmetically Cohen-Macaulayness ofthe variety among others characteristics.

In a more general context, this problem was solved by Lascoux in [16] where he gavea minimal free resolution of Schubert’s varieties. His construction rests heavily on thetheory of Schur functors and the fact that, in characteristic zero, the Schur functors are theirreducible representations of the general linear group. Despite the fact that the techniquesdeveloped by Lascoux are out of reach for an undergraduate student of Mathematics,the particular case of determinantal varieties admits a treatment based on developingsuitable rudiments of multilinear algebra and combinatorics. In this sense, this paper isan approachment to the work of Lascoux in order to study a minimal free resolution ofthe ideal associated to the minors of a matrix. Our main goal is to construct a minimal freecomplex of these ideals whose modules are the modules of the minimal free resolutiongiven by Lascoux, and next describe the resolution. In fact, this minimal free complex is agood candidate to be a resolution, too.

This paper is organized as follows. In Chapter 1 we present a review of backgroundmaterial on Hopf algebras, including exterior, symmetric and divided power algebras, andan exposition of partitions, Tableaux and Young diagrams which are the basic tools usedalong the main body of the paper. Chapter 2 deals with Schur and CoSchur functors onfree R-modules, where R is a commutative ring of arbitrary characteristic. In section 2.1we define the Schur functor Lλ/µF of a free R-module F with respect to the skew partitionλ/µ as the image of a natural transformation dλ/µ : Λλ/µF → Sλ̃/µ̃F of free R-modules. In2.1.1 the freeness of Schur functors is proved giving an explicit basis of Lλ/µF through abasis of F which is described by means of Young tableaux. We end this subsection givingthe rank of LλF and with a discussion about the functoriality of Lλ/µ(−). The CoSchurfunctor Kλ/µF has a similar treatment in section 2.2, we give a basis of Kλ/µF and we showthe duality (Lλ/µF)∗ � Kλ̃/µ̃F

∗, where F∗ denote the dual of F.The remainder of Chapter 2 is devoted to the properties of Schur functors. In sec-

tion 2.3, we provide a natural filtration of Lλ/µ(F ⊕ G) with associated graded object⊕µ⊆γ⊆λ Lγ/µF⊗ Lλ/γG, where F and G are free R-modules. In characteristic zero this as-

sociated graded object becomes a direct sum decomposition of the Schur functor Lλ/µ(F⊕G). In section 2.4, we address to the Cauchy formula for exterior and symmetric alge-bra. Sk(F ⊗ G) and Λk(F ⊗ G) have natural filtrations with associated graded objects⊕|λ|=k LλF ⊗ LλG and

⊕|λ|=k LλF ⊗ KλG, respectively. In characteristic zero, this sum

becomes a direct decomposition of these functors. The Cauchy formula for exterior alge-bra is one of the pillars in the Lascoux construction. Finally, section 2.5 is devoted to the

iv Introduction

Littlewood-Richardson rule for Schur functors. More precisely, when R contains a fieldof characteristic zero, the tensor product LλF ⊗ LµF of Schur functors decomposes in adirect sum ∑ν u(λ, µ; ν)LνF, where u(λ, µ; ν) are the multiplicities of the factor LνF givenby the Littlewood-Richardson rule. As a consequence, we obtain the Pieri formulas forSchur functors which are essential to describe the boundary maps of the resolution givenby Lascoux.

In Chapter 3, we construct a minimal free complex of the ideal generated by the minorsof a matrix and a minimal free resolution of determinantal ideals. In section 3.1, weassume that R is a local commutative ring containing a field K of characteristic zero withmits maximal ideal, F and G are free R-modules of ranks m+ t− 1 and n+ t− 1 respectivelyand (ri,j) is a matrix of size (m + t− 1)× (n + t− 1) with entries in R associated to a R-map Φ : F → G∗. We denote by It the ideal generated by the t× t minors of (ri,j) and wedefine a complex (C•(Φ, t), d•) of length mn as follows:

(i) For all k, Ck(Φ, t) is a free R-module, it is a direct sum of tensor product of Schurfunctors.

(ii) d1(C1(Φ, t)) = It and dk(Ck(Φ, t)) ⊂ mCk−1(Φ, t).

We construct the modules Ck(Φ, t) by means of the action of the group algebra K(Sk) onthe tensor products TkF and TkG. In subsection 3.1.1 we provide all the results requiredon K(Sk) and Young tableaux. In 3.1.2 we construct the modules Ck(Φ, t) as the imagesof the morphisms associated to idempotents of K(Sk). The boundary maps dk are definedin 3.1.2 and they are essentially the maps induced by the contractions TkF ⊗ TkG → Rextending Φ. The remainder of the section 3.1 is devoted to prove that (C•(Φ, t), d•) is acomplex. This rests on two formulas involving the idempotents defining the modules ofthe complex.

In section 3.2, assuming notations in 3.1 with R = K[x0, . . . , xs] we define determinantalideals and determinantal varieties. In 3.2.1 we present the minimal free resolution (L•,t, d•)of determinantal ideals. The free R-modules Lk,t are

⊕|I|+n(I)=k,I′,0 L Ĩ F ⊗ L Ĩ′G, where I

is a partition of weight m + k− 1, I′ and n(I) are described by means of I and its Durfeesquare. In fact, Lk,t = Ck,t(Φ, t). The boundary maps dk are given by Pieri formulas andthe natural contractions ΛρF ⊗ ΛρG → R induced by Φ. The remains of the Chapterdeals with examples of minimal free resolutions and two explicit resolutions computedwith the program Macaulay2 [10]. The Eagon-Northcott complex gives a minimal freeresolution of determinantal ideals generated by the maximal minors of a homogeneousmatrix. We explain this resolution using 3.2.1, but actually the original treatment of thisproblem due to Eagon-Northcott in [7] does not involve Schur functors and this particularcase gives a resolution of determinantal ideals even when R is not of characteristic zero.The Gulliksen-Negard complex is a minimal free resolution of the determinantal idealgenerated by the submaximal minors of a square matrix. This complex has only foursubmodules and boundary maps, we do an accurate description of them.

Introduction v

Acknowledgements.

I would like to express my infinite gratitude to my advisor Rosa Maria Miró-Roig forher dedication and effort week after week to get this work to succeed in. I would liketo thank also to P. C. Roberts who kindly sent to us his preprint which was essential todevelop this paper. And finally, I would like to thank to my family for their support.

To M.

vi Introduction

Chapter 1

Preliminaries.

This chapter is a compilation of all background material on multilinear algebra andcombinatorial which is utilized in the main body of the paper. Hopf algebras and discus-sions of the relevant properties of the exterior, symmetric and divided power algebras areincluded. We follow [15] and [20] in section 1.1, and [1] in section 1.2.

1.1 Multilinear algebra.

1.1.1 Hopf algebras.

Let R be a commutative ring.

Definition 1.1.1. Let M and N be two graded R-modules. We define the twisting mor-phism T : M⊗R N → N ⊗R M by T(x⊗ y) = (−1)ijy⊗ x, x ∈ Mi, y ∈ Nj.

Definition 1.1.2. A graded R-Hopf algebra is a graded R-module A = ∑i≥0 Ai together witha multiplication m : A⊗R A→ A, a unit η : R→ A, a comultiplication or diagonalization∆ : A→ A⊗ A and a counit ε : A→ R satisfying:

1. (A, m, η) is a graded R-algebra, (A, ∆, ε) is a graded R-coalgebra, the counit ε is anR-algebra map and the unit η is an R-coalgebra map. That is, the following diagramsare commutative:

A A⊗ A

A⊗ A A⊗ A⊗ A

∆

∆

∆⊗ IdId⊗ ∆

A

R⊗ A A⊗ A A⊗ R

� �

ε⊗ Id Id⊗ ε.

2. The multiplication m and the comultiplication ∆ are compatible in the sense that thefollowing diagram commutes:

1

2 Preliminaries.

A⊗ A A A⊗ A

A⊗ A⊗ A⊗ A A⊗ A⊗ A⊗ A

∆⊗ ∆

m ∆

Id⊗ T ⊗ Idm⊗m

.

In addition, we say that A is a commutative graded R-Hopf algebra if the followingdiagrams commute,

A⊗ A A⊗ A

A

T

m m

A

A⊗ A A⊗ A

∆ ∆

T .We assume that A is connected (i.e. A0 = R) and free (i.e. each i ≥ 0, Ai is a finitely

generated free R-module).

Definition 1.1.3. Let (A, mA, ηA) and (B, mB, ηB) be two graded R-Hopf algebras. We saythat an R-map α : A → B is a map of graded R-Hopf algebras if mB ◦ (α⊗ α) = α⊗mA,and ∆B ◦ α = (α⊗ α) ◦ ∆A.

Definition 1.1.4. Let A = (A, mA, ηA, ∆A, εA) and (B, mB, ηB, ∆B, εB) be two graded R-Hopf algebras. The tensor product A ⊗ B is defined by the graded R-module A ⊗ B =∑i≥0 Ai ⊗R Bi, a multiplication mA⊗B : A ⊗ B ⊗ A ⊗ B

Id⊗T⊗Id−−−−−→ A ⊗ A ⊗ B ⊗ B mA⊗mB−−−−→A⊗ B, a comultiplication ∆A⊗B : A⊗ B

∆A⊗∆B−−−−→ A⊗ A⊗ B⊗ B Id⊗T⊗Id−−−−−→ A⊗ B⊗ A⊗ B, aunit ηA⊗B : R

ηA⊗ηB−−−−→ A⊗ B and a counit εA⊗B : A⊗ BεA ·εB−−−→ R.

1.1.2 The Exterior Algebra.

Let R be a commutative ring and let F be a free R-module of rank n with an orderedbasis {x1, . . . , xn}. We denote F∗ = Hom(F, R) the dual of F with basis {x∗1 , . . . , x∗n} dualto the basis {x1, . . . , xn}. We denote by Sr the set of all permutations of {1, . . . , r}.

Definition 1.1.5. Let Λ0F = R and Λ1F = F. For all integer r > 1 we define the r-th exteriorpower ΛrF of F to be quotient of the r-th tensor power TiF := F⊗ · · · ⊗ F of F respect to thesubmodule E of TrF generated by the elements f1 ⊗ · · · ⊗ fr − (−1)sg(σ) fσ(1) ⊗ · · · ⊗ fσ(r)for all σ ∈ Sr and f1, . . . , fr ∈ F. In other words, ΛrF is the image by the natural surjectivemap π : TrF → TrF/E and then, the generators of ΛrF are the elements π( f1 ⊗ · · · ⊗ fr)such that f1, . . . , fr ∈ F which we denote by f1 ∧ · · · ∧ fr.

Let f1 ∧ · · · ∧ fr ∈ ΛrF. From the above definition it follows that for each i , j ∈{1, . . . , r}, f1 ∧ · · · ∧ fi ∧ · · · ∧ f j ∧ · · · ∧ fr = − f1 ∧ · · · ∧ f j ∧ · · · ∧ fi ∧ · · · ∧ fr. So, it isclear that if there are two indices such that fi = f j, then f1 ∧ · · · ∧ fr = 0. We assume thatin the expression f1 ∧ · · · ∧ fr all elements fi are different, otherwise it is 0.

Proposition 1.1.6. The set {xi1 ∧ · · · ∧ xir | 1 ≤ i1 < · · · < ir ≤ n} form a basis of ΛrF. In

particular ΛrF is a free R-module of rank (nr).

1.1 Multilinear algebra. 3

The r-th exterior power ΛrF is a submodule of TrF through the natural immersionı : ΛrF → TrF defined by ı( f1 ∧ · · · ∧ fr) = ∑σ∈Sr (−1)

sg(σ) fσ(1) ⊗ · · · ⊗ fσ(r).

Proposition 1.1.7. Let F and E be two free R-modules. If φ : E → F is an R-map, thenwe have a well-defined linear map Λrφ : ΛrE → ΛrF defined by Λrφ(e1 ∧ · · · ∧ er) =φ(e1) ∧ · · · ∧ φ(er). Thus, the r-th exterior power is an endofunctor on the category ofR-modules and R-maps.

Definition 1.1.8. We define ΛF to be the graded R-module ΛF :=⊕

r≥0 ΛrF.

For each s, t ≥ 0 there are natural maps mr,s : ΛrF⊗ΛsF → Λr+sF and ∆r+s : Λr+sF →ΛrF⊗ΛsF given by the formulas

mr,s( f1 ∧ · · · ∧ fr ⊗ g1 ∧ · · · ∧ gs) = f1 ∧ · · · ∧ fr ∧ g1 ∧ · · · ∧ gs

∆r+s( f1 ∧ · · · ∧ fr+s) = ∑σ∈Sr+s

(−1)sg(σ) fσ(1) ∧ · · · ∧ fσ(r) ⊗ fσ(r+1) ∧ · · · ∧ fσ(r+s)

respectively, where σ is such that σ(1) < · · · < σ(r) and σ(r + 1) < · · · < σ(r + s).These maps induce a natural multiplication m : ΛF ⊗ ΛF → ΛF and comultiplication∆ : ΛF → ΛF⊗ΛF in ΛF given by all components mr,s and ∆r+s respectively. Then, wehave

Proposition 1.1.9.

(i) (ΛF, m̃, η, ∆, ε) is a connected, free and commutative graded R-Hopf algebra wherethe unit is the natural inclusion η : R → ΛF into degree 0 and the counit is theprojection ε : ΛF → R into degree 0.

(ii) The component ∆r+s of the comultiplication map in ΛF∗ is the dual map of thecomponent of the multiplication map mr,s in ΛF.

(iii) The component mr,s of the multiplication map in ΛF∗ is the dual of the componentof the comultiplication map ∆r+s in ΛF.

1.1.3 The Symmetric Algebra.

Definition 1.1.10. Let S0F = R and S1F = F. For each r > 1 we define the r-th symmetricpower of F to be the quotient of TrF respect to the submodule S of TrF generated by theelements f1 ⊗ · · · ⊗ fr − fσ(1) ⊗ · · · ⊗ fσ(r) for all σ ∈ Sr and f1, . . . , fr ∈ F. Then SrF is theimage under the natural projection π : TrF → TrF/S and it is generated by the elementsπ( f1 ⊗ · · · ⊗ fr) := f1 · · · · · fr, f1, . . . , fr ∈ F.

Observe that the above definition differs from ΛrF only by a sign, (−1)sg(σ), howeverthis detail makes important distinctions. Let f1 · · · · · fr ∈ SrF. For each i , j ∈ {1, . . . , r}we have f1 · · · · · fi · · · · · f j · · · · · fr = f1 · · · · · f j · · · · · fi · · · · · fr . Differs from ΛrF, inthe expressions f1 · · · · · fr we can have two equals elements. In this sense, the elementf1 · · · · · fr could not have a minimal expression. To emphasize this fact we will use thenotation f ir1 · · · · · f

irt such that i1 + · · ·+ ir = r where we assume that all elements fi are

different and eventually ik = 0 with fiki = 1 ∈ R.

4 Preliminaries.

Proposition 1.1.11. {xi11 · · · · · xinn | i1 + · · ·+ in = r} form a basis of SrF. In particular,

SrF is a free R-module of rank (n+r−1r ).

SrF is a submodule of TrF through the natural immersion : SrF → TrF defined by( f1 · · · · · fr) = ∑σ∈Sr fσ(1) ⊗ · · · ⊗ fσ(r).

Proposition 1.1.12. Let F and E be two free R-modules and let φ : F → E be an R-map.Then, the map Srφ : SrE → SrF defined by Sr(φ)(e1 · · · · · er) = φ(e1) · · · · · φ(er) is awell-defined R-map. In other words, the rth symmetric power defines an endofunctor onthe category of free R-modules and R-maps.

Definition 1.1.13. We define SF to be the graded R-module SF :=⊕

r≥0 SrF.

As we have seen in ΛF, there are naturals maps mr,s : SrF ⊗ SsF → Sr+s and ∆r+s :Sr+sF → SrF⊗ SsF defined by the formulas

mr,s( f1 · · · · · fr ⊗ g1 · · · · · gr) = f1 · · · · · fr · g1 · · · · · gs

∆r+s( f1 · · · · · fr+s) = fσ(1) · · · · · fσ(r) ⊗ fσ(r+1) · · · · · fσ(r+s)

respectively where σ runs over the set of all permutations of {1, . . . , r + s} such thatσ(1) < · · · < σ(r), σ(r + 1) < · · · < σ(r + s), which induce a multiplication m and ∆comultiplication in SF. In particular, ∆r+s( f

i11 · · · · · f

ir+sr+s ) = ∑0≤jk≤ik (

i1j1) · · · (ir+sjr+s) f

j11 · · · · ·

f jr+sr+s ⊗ fi1−j11 · · · · · f

ir+s−jr+sr , where jk = 0 if ik = 0 and j1 + · · ·+ jr+s = r.

Proposition 1.1.14. (SF, m, η, ∆, ε) is a connected, free and commutative graded R-Hopfalgebra with the obvious unit η : R→ SF and counit ε : SF → R.

1.1.4 The Divided Power Algebra.

Definition 1.1.15. Let D0F = R and D1F = F. For each r > 1 we define the rth dividedpowerDrF of F to be the dual of the rth symmetric power SrF∗.

Considering the basis {(x∗1)i1 · · · · · (x∗n)in | i1 + · · · + in = r} of Sr(F∗), we definex(i1)1 · · · · · x

(in)n to be the dual element of the basis element (x∗1)

i1 · · · · · (x∗n)in . Then,

Proposition 1.1.16. {x(i1)1 · · · · · x(in)n | i1 + · · ·+ in = r} form a basis of DrF. In particular,

DrF is a free R-module of rank (n+r−1r ).

Proposition 1.1.17. DF =⊕

r≥0 DrF is a graded dual of the symmetric algebra SF∗ withthe obvious unit and counit and where

(i) The component mr,s : DrF ⊗ DsF → Dr+sF of the multiplication on DF is the dualof the component ∆r+s : Sr+sF∗ → SrF∗ ⊗ SsF∗ of the comultiplication on SF∗.

mr,s is given by mr,s(x(i1)1 · · · · · x

(in)n ⊗ x

(j1)1 · · · · · x

(jn)n ) = (

i1+j1j1

) · · · (in+jnjn )x(i1+j1)1 · · · · ·

x(in+jn)n .

1.2 Combinatorics. 5

(ii) The component ∆r+s : Dr+sF → DrF⊗DsF of the diagonalization on DF is the dualof the component mr,s : SrF∗ ⊗ SsF∗ → Sr+sF∗ of the multiplication on SF∗.

∆r+s is given by ∆r+s(x(i1)1 · · · · · x

(in)n ) = ∑0≤jk≤ik e

(j1)1 · · · · · e

(jn)n ⊗ e

(i1−j1)1 · · · · · e

(in−jn)n

where j1 + · · ·+ jn = r.

Definition 1.1.18. Let f ∈ F with f = ∑ni=1 uixi. Let f (0) = 1 ∈ R and f (1) = f . For eachr > 1 we define the r-th divided power f (r) ∈ DrF of f to be ∑i1+···+in=r u

i11 · · · u

inn x

(i1)1 · · ·

x(in)n .

Proposition 1.1.19. Let f , g ∈ F and let p, q be two non negative integers. The dividedpowers have the following properties:

1. f (p) f (q) = (p+qq ) f(p+q) ∈ Dp+qF

2. ( f + g)(p) = ∑pk=0 f

(k)g(p−k).

3. ( f g)(p) = f (p)g(p).

4. ( f (p))(q) = (pq)!q!pq! f(pq).

Proposition 1.1.20. Let F and E be two free R-modules and let φ : F → E be an R-map.We denote by φ∗ : E∗ → F∗ the dual map of φ. Then, we have a well defined R-mapDrφ : DrF → DrE which is the dual map of the map Sr(φ∗) : SrE∗ → SrF∗. Moreprecisely, the rth exterior power defines an endofunctor on the category of free R-modulesand R-maps.

1.2 Combinatorics.

1.2.1 Partitions and skew partitions.

Let N∞ := {λ := {λi}i≥1 | λi = 0 but a finite number of terms}. In other words,for each non negative integer p, let Np := {(λ1, . . . , λp) | λi ∈ N, i = 1, . . . , p}. HenceN∞ = ∪p≥1Np.

We will consider (λ1, . . . , λp) and {λ1, . . . , λp, 0, . . .} the same element.

Definition 1.2.1. A partition is an element λ = {λi}i≥1 ∈ N∞ whose components arearranged in decreasing order, that is λi ≥ λi+1, ∀i ≥ 0. The length of a partition λ ∈ N∞is the number of non zero components in the partition, we often denote it by l(λ). Theweight of a partition λ ∈N∞ is the sum of all its components. We note it by |λ| := ∑i≥1 λiand we will say that λ is a partition of weight |λ|.

For example, (10, 8, 9, 11, 4, 3, 2) is not a partition and (10, 7, 3, 2) is a partition of weight22 and length 4.

Definition 1.2.2. Let λ = (λ1, . . . , λp) ∈N∞ be a partition. For each i ∈ {1, . . . , λ1}, let λ̃ibe the number of terms of λ which are greater than or equal to i. Clearly λ̃i ≥ λ̃i+1, ∀i ∈{1, . . . , λ1}. The transpose or conjugate of λ is the partition λ̃ = (λ̃1, . . . , λ̃λ1).

6 Preliminaries.

For example, the transpose of (10, 7, 3, 2) is the partition (4, 4, 3, 2, 2, 2, 2, 1, 1, 1).

Remark 1.2.3. We consider that all terms of a partition λ = (λ1, . . . , λp) are non zero.Note that hence, the length of λ is p and equals to λ̃1.

Definition 1.2.4. Let λ = (λ1, . . . , λp) ∈ N∞ be a partition. The diagram or shape of λ is∆λ := {(i, j) ∈ N2 | 1 ≤ i ≤ p, 1 ≤ j ≤ λi}. We will represent the graphic of ∆λ in N2,where the points (i, j) ∈ ∆λ will be represented by squares.

For example, ∆(6,4,3,2), ∆(4,4,3,2,1,1) and ∆(5,4,3,2,1) correspond to

respectively.We can consider the shape of a partition λ = (λ1, . . . , λp) as a kind of matrix with

p rows with different lengths and λ1 columns. The ith row of ∆λ is of length λi. Now,the jth column of ∆λ is of length the number of terms in λ greater than or equal to j,which equals λ̃j. In fact, ∆λ̃ is the diagram transpose of ∆λ, and hence this tell us that the

conjugate of λ̃ is λ. That is, ˜̃λ = λ and we can conclude that the conjugation λ → λ̃ is aninvolution on the set of partitions.

The notions of partitions are, in fact, an introduction of a more generalized concept weexplain next. In the set of partitions one can define an order. Let λ = (λ1, . . . , λp) andµ = (µ1, . . . , µq) be two partitions, we say µ ⊆ λ if p ≥ q and µi ≤ λi, ∀i ∈ {1, . . . , q}. It isequivalent to say that the diagram of µ is contained in the diagram of µ, that is ∆µ ⊆ ∆λ.

Definition 1.2.5. Let µ = (µ1, . . . , µq) and λ = (λ1, . . . , λp) be two partitions such thatµ ⊆ λ. We define the skew partition λ/µ := (λ1 − µ1, . . . , λq − µq, λq+1, . . . , λp) with skewshape of diagram ∆λ/µ := ∆λ − ∆µ = {(i, j) ∈N2 | 1 ≤ i ≤ p; µi + 1 ≤ j ≤ λi}.

For convenience we will write µ = (µ1, . . . , µp) where µq+1 = · · · = µp = 0.For example, the skew diagrams associated to (4, 3, 2)/(2, 1), (5, 2, 1)/(4, 0, 0) and(7, 6, 3)/(4, 5, 1) correspond to

Note that, the skew partition λ/µ is a partition if, and only if the length of µ equalsthe length of λ or µ is the zero partition. Clearly, the set of partitions is a subset of the setof skew partitions, indeed λ/(0) = λ.

1.2 Combinatorics. 7

1.2.2 Tableaux.

Definition 1.2.6. Let λ/µ a skew partition and let S be a totally ordered set. A tableauof shape λ/µ with values in the set S is a function from the skew shape ∆λ/µ to S. Wedenote Tabλ/µ(S) the set of such tableaux.

We will represent graphically T ∈ Tabλ/µ(S) by the skew shape ∆λ/µ filled with ele-ments of S.

Let us see few simple examples. Consider S = {1, . . . , 5} with the usual order on Nand λ = (3, 2). Then, T : ∆λ → S such that T(i, j) = max{i, j} is a tableau of shape λ withvalues in S. Two more examples: R : ∆λ → S such that R(i, j) = i + j and U : ∆λ → Ssuch that U(i, j) = i.

T = 1 2 32 2

R = 2 3 43 4

U = 1 1 12 2

Definition 1.2.7. Let T ∈ Tabλ/µ(S). We say T is row-standard if T(i, j) < T(i, j + 1)for all (i, j), (i, j + 1) ∈ ∆λ/µ. We say T is column-standard if T(i, j) ≤ T(i + 1, j) for all(i, j), (i + 1, j) ∈ ∆λ/µ. Finally, T is called standard if it is row and column standard.

Continuing with the examples above, T(i, j) = max(i, j) and U(i, j) = i are bothcolumn-standard but not row-standard while R(i, j) = i + j is standard.

Definition 1.2.8. Let T ∈ Tabλ/µ(S). We say that T is co-row-standard if T(i, j) ≤ T(i, j +1) ∀(i, j), (i, j + 1) ∈ ∆λ/µ. We say that T is co-column-standard if T(i, j) < T(i + 1, j) ∀(i, j),(i + 1, j) ∈ ∆λ/µ. Finally, T is co-standard if is co-row and co-column standard.

Observe that T is co-row-standard but not co-column-standard, while R and U areboth co-standard.

8 Preliminaries.

Chapter 2

Schur Functors and CoSchurFunctors.

This chapter is devoted to an introduction of Schur and CoSchur functors theory usingonly elementary rudiments of multilinear algebra and combinatorics, which we have justpresented into the Introductory Material. We follow basically the contents from [1] witha few references from [20] and [9]. In sections 2.1 and 2.2 we define Schur and CoSchurfunctors and we establish the freeness of both functors. In the remaining sections wegive decomposition formulas for Schur functors including Littlewood-Richardson rule andCauchy formulas.

2.1 Schur Functors.

We start with some notations we will use along this paper. Let R be a commutativering and let F be a free R-module of rank n. Let µ = (µ1, . . . , µp) and (λ1, . . . , λp) be twopartitions such that µ ⊆ λ. We define the free R-modules

Λλ/µF := Λλ1−µ1 F⊗R · · · ⊗R Λλp−µp F

Sλ/µF := Sλ1−µ1 F⊗ · · · ⊗ Sλp−µp F

Dλ/µF := Dλ1−µ1 F⊗ · · · ⊗ Dλp−µp F

Tλ/µF := F(1,µ1+1) ⊗ · · · ⊗ F(1,λ1) ⊗ · · · ⊗ F(p,µp+1) ⊗ · · · ⊗ F(p,λp) =⊗

(i,j)∈∆λ/µ F(i,j), whereF(i,j) denotes a copy of F and the tensor power ⊗ is over R. Note that Tλ/µF is an usualway to write the tensor power T|λ/µ|F where |λ/µ| = |λ| − |µ|.

Definition 2.1.1. We define the Schur map dλ/µ : Λλ/µF → Sλ̃/µ̃F to be the composition

Λλ/µFα−→ Tλ/µ

β−→ Sλ̃/µ̃F,

9

10 Schur Functors and CoSchur Functors.

where α is the tensor product of the natural inclusions or diagonalizations ıi : Λλi−µi F →Tλi−µi F = F(i,µi+1) ⊗ · · · ⊗ F(i,λi), i = 1, . . . p, and β is the tensor product of the multi-plications mj : F(µ̃j+1,j) ⊗R · · · ⊗R F(λ̃j ,j) → S

λ̃j−µ̃j F, j = 1, . . . , λ1. Note that ı is a tensorproduct of appropriated comultiplications on ΛF. We will often denote ıi simply by ∆.

There is another way to define the Schur map which we will often use. It is a simplechange of notation using Ferrers matrices which sometimes makes operations easier. Wewill use both indifferently.

The Ferrers matrix associated to the skew partition λ/µ is a λ1 × λ1 matrix of zerosand ones αλ/µ = (ai,j) defined by{

ai,j = 1 i f µi + 1 ≤ j ≤ λiai,j = 0 i f 1 ≤ j ≤ µi or λj+1 ≤ j ≤ λ1

Keeping in mind that Fai,j = R i f ai,j = 0, dλ/µ is defined as the composition Λλ/µF →⊗ai,j∈αλ/µ Fai,j → Sλ̃/µ̃F, where Fai,j is a copy of F, the first map is the tensor product of

the natural injections Λλi−µi F → Fai,1 ⊗ · · · ⊗ Fai,λ1 , i = 1, . . . , q, and the second map isthe tensor product of the multiplication Fa1,j ⊗ · · · ⊗ Faq,j → S

a1,j+···+aq,j F = Sλ̃j−µ̃j F, j =1, . . . , λ1, since a1,j + · · ·+ aq,j = λ̃j − µ̃j.

In fact, we can generalize the above map to any matrix of zeros and ones.Let α = (αi,j) be an arbitrary s × t matrix of zeros and ones. For each i = 1, . . . , s

let pi = ∑tj=1 ai,j and for each j = 1, . . . , t let qj = ∑si=1 ai,j. Then we can consider free

R-modulesΛαF := Λp1 F⊗ · · · ⊗Λps F and Sα̃F := Sq1 F⊗ · · · ⊗ Sqt F

and we can define a natural map dα : ΛαF → Sα̃F in the same manner as the Schur map.

Definition 2.1.2. The image of dλ/µ is called the Schur functor of F with respect to theskew partition λ/µ, and it is denoted by Lλ/µF.

Let us see some examples.

Examples 2.1.3. (1) If λ = (m), then λ̃ = (1, . . . , 1) and the Schur map d(m) : Λ(m)F →S(1,...,1)F is just the natural inclusion of ΛmF on TmF. In this case, we see thatL(m) � ΛmF.

(2) If λ = (1, . . . , 1), then λ̃ = (m), and the Schur map d(1,...,1) : Λ(1,...,1)F → S(m)F is justthe multiplication F⊗ · · · ⊗ F → SmF. Clearly, L(1,...,1)F is the mth symmetric powerSmF.

(3) Let λ = (3, 2), first we describe the Schur map associated with λ.

Remember⊗

(i,j)∈∆λ F(i,j) = T|λ|F. We have,

d(3,2) : Λ3F⊗R Λ2F →

⊗(i,j)∈∆λ

F(i,j) = T5F → S2F⊗R S2F⊗R F

2.1 Schur Functors. 11

u ∧ v ∧ w⊗ x ∧ y→u⊗ v⊗w⊗ x⊗ y− u⊗ v⊗w⊗ y⊗ x− u⊗w⊗ v⊗ x⊗ y + u⊗w⊗ v⊗ y⊗ x −v⊗u⊗w⊗ x⊗ y + v⊗ u⊗w⊗ y⊗ x + v⊗w⊗ u⊗ x⊗ y− v⊗w⊗ u⊗ y⊗ x +w⊗ u⊗v⊗ x⊗ y− w⊗ u⊗ v⊗ y⊗ x− w⊗ v⊗ u⊗ x⊗ y + w⊗ v⊗ u⊗ y⊗ x→ ux ⊗ vy ⊗ w − uy ⊗ vx ⊗ w − ux ⊗ wy ⊗ v + uy ⊗ wx ⊗ v − vx ⊗ uy ⊗ w + vy ⊗ux⊗ w +vx⊗ wy⊗ v− vy⊗ wx⊗ v + wx⊗ uy⊗ v− wy⊗ ux⊗ v− wx⊗ vy⊗ u +wy⊗ vx⊗ u.

It is clear that Lλ/µF is a submodule of Sλ̃/µ̃F, but it is not trivial that actually Lλ/µFis a free R-module.

2.1.1 The freeness of Schur Functors.

As we have anticipated in the previous subsection, the modules Lλ/µF are free. Wewill show this fact by finding a basis to Lλ/µF through a basis of F. However, first weneed to solve the computational problem of dλ/µ.

Let λ/µ be a skew partition with λ̃1 = q and let BF := {x1, . . . , xn} be a basis of F. Wecan describe a basis of Λλ/µF and Sλ/µF as follows.

For each i ∈ {1, . . . , q} we denote by Ii = {α(i,µi+1), . . . , α(i,λi)} a strictly increasingsubset of {1, . . . , n} such that α1 < . . . < αs. Clearly, the elements XI1 ⊗ · · · ⊗ XIq , whereeach Ii is a such of these subsets, form a basis of Λλ/µF.

Now, if we choose S to be BF with the order x1 < · · · < xn, then for each basis elementX = XI1 ⊗ · · · ⊗XIq we can associate a tableau TX ∈ Tabλ/µ(S) defined by TX(i, j) = Xα(i,j) .Clearly, TX is a row-standard tableau. And conversely, if T ∈ Tabλ/µ(S) is a row-standardtableau, then the element XT := XI1 ⊗ · · · ⊗ XIq where XIi = T(i, µi+1) ∧ · · · ∧ T(i, λi) is asuch of basis element we have just described. So, {XT | T ∈ Tabλ/µ(S) is row-standard}is a basis of Λλ/µF.

For example, let λ = (4, 2, 1) and n = 10. Then, the tableau

T = x1 x4 x5 x7x2 x10x3

define the basis element XT = x1 ∧ x4 ∧ x5 ∧ x7 ⊗ x2 ∧ x10 ⊗ x3 in Λ(4,2,1)F.In the same way, we can describe a basis of Sλ̃/µ̃F through Tabλ/µ(S). If T ∈ Tabλ/µ{x1,

. . . , xn} is column-standard, then we can associate to it the element ZT = XJ1 ⊗ · · · ⊗ XJλ1 ,where XJi = T(µ̃i, i) · · · · · T(λ̃i, i). Clearly, {ZT | T ∈ Tabλ/µ(S) is column-standard} is abasis of Sλ̃/µ̃F.

For example, let λ = (4, 3, 2, 1) and n = 5. The tableau

T = x1 x3 x4 x5x1 x5 x5x1 x5x2

define the basis element ZT = x31x2 ⊗ x3x25 ⊗ x4x5 ⊗ x5 of S(4,3,2,1)F.


From this follows that {dλ/µ(XT) | T ∈ Tabλ/µ(S) is row-standard} generate Lλ/µF.We will show that {dλ/µ(XT) | T ∈ Tabλ/µ(S) is standard} is a basis of Lλ/µF.

Let T ∈ Tabλ/µ(S) with λ = (λ1, . . . , λq) and µ = (µ1, . . . , µq). Remember dλ/µ =(m1⊗ · · · ⊗mλ1)(ı1⊗ · · · ⊗ ıq). Thus, (ı1⊗ · · · ⊗ ıq)(XT) = ∑σ=(σ1,...,σq)(−1)

sg(σ)XTσ whereσi is a permutation of 1, . . . , λi − µi and Tσ is the tableau defined by Tσ(i, j) = T(i, σi(j)).Applying (m1 ⊗ · · · ⊗ mλ1) we obtain dλ/µ(XT) = ∑σ(−1)

sg(σ)ZTσ , where ZTσ ∈ Sλ̃/µ̃Fsimilarly as we have seen before.

At this moment we will introduce some facts about tableaux which we will need next.Let S = BF.

Definition 2.1.4. Let T ∈ Tabλ/µ(S) and let p, q be positive integers. We define Tp,q tobe the number of times the elements x1, . . . , xq appear as entries in the first p rows of T.Formally, Tp,q = |{(i, j) ∈ T | i ≤ p and T(i, j) ∈ {x1, . . . , xq}}|.

For example,T = x1 x4 x5 x7

x2 x10x3

, T5,2 = 4.

Definition 2.1.5. Let T, S ∈ Tabλ/µ(S). We say S ≤ T if Sp,q ≥ Tp,q for every positiveintegers p, q and we say S < T if S ≤ T and Sp,q > Tp,q for at least one pair p, q.

Proposition 2.1.6. ≤ is a reflexive and transitive relation on Tabλ/µ(S).Proof. Obviously ≤ is reflexive, Tp,q = Tp,q for every p, q, so T ≤ T. Let T, W, R ∈Tabλ/µ(S) such that T ≤ W and W ≤ R, we want to see T ≤ R. Let p, q be positiveintegers, by hypothesis Rp,q > Wp,q > Tp,q and hence Rp,q > Tp,q. �

Remark 2.1.7. If we restrict ≤ to the subset of row-standard tableaux, then ≤ is consis-tent with the lexicographic order induced by the correspondence between row-standardtableaux and the basis elements of Λλ/µF.

Lemma 2.1.8. Let T, R ∈ Tabλ/µ(S) where R is formed by exchanging certain entries fromthe kth row of T, say T(k, l1), . . . , T(k, la), to certain entries of the (k + 1)th row of T, sayT(k + 1, m1), . . . , T(k + 1, ma), where T(k + 1, mi) < T(k, li) for i = 1, . . . , a. Then R < T.Proof. First we consider the simple case a = 1. We have , R(k, l) = T(k + 1, m), R(k +1, m) = T(k, l) and R(i, j) = T(i, j) for all (i, j) , (k, m), (k + 1, m). By hypothesis R(k, l) =T(k + 1, m) < T(k, l) = R(k + 1, m).

Let p, q be positive integers, remember Rp,q = |{(i, j) ∈ S | i ≤ p and S(i, j) ∈{x1, . . . , xq}|. Clearly Rp,q = Tp,q if p < k since R equals to T at the first k− 1 rows. We fixp = k, R differs T only at the entry (k, l), and R(k, l) appears at the first k rows of R onemore time than of T, respectively S(k + 1, m) one less. Clearly Rk,q = Tk,q i f xq < R(k, l),or xq ≥ R(k + 1, m), since R(k, l) < R(k + 1, m) and Rk,q = Tk,q + 1− 1. If R(k, l) ≤ xq <R(k + 1, m), then Rk,q = Tk,q + 1. Since Rp,q equals Tp,q if p > k, we conclude R < T.

Considering that we can see R as a result of a series of tableau Ri formed by exchangingRi(k, li) = Ri−1(k + 1, mi) and Ri(k + 1, mi) = Ri−1(k, li) with 1 ≤ i ≤ a and R1 = T andhence Ri−1 < Ri as we have seen at case a = 1, the general case follows because of thetransitivity of ≤. �


Our first goal will be to show that B = {dλ/µ(XT) | T ∈ Tabλ/µ(S) is standard} is asystem of generators of Lλ/µF. We will proceed defining an appropriate map whose imageis contained in the kernel of dλ/µ and which will allow us to prove that dλ/µ(XT), withT row-standard, is a linear combination of elements of B. Moreover, we will establish anisomorphism between the cokernel of this map and Lλ/µF. This fact will give us a naturalway to describe the modules Lλ/µF.

Lemma 2.1.9. For each i ∈ {1, . . . , q− 1}, the map dλ/µ can be factored as follows:Λλ/µF

ρ−→ Λλ1−µ1 F⊗ · · · ⊗Λλi−1−µi−1 F⊗ Sai,1+ai+1,1 F⊗ · · · ⊗ Sai,λ1+ai+1,λ1 F⊗Λλi+2−µi+2 F

⊗ · · · ⊗ Λλq−µq F η−→ ⊕ak,j∈α,ki+1 F(k,j)ν−→ Sλ̃/µ̃F, where ρ = Id⊗ · · · ⊗ Id⊗ d(λi ,λi+1)/(µi ,µi+1) ⊗ Id⊗ · · · ⊗ Id, η = ı1 ⊗ · · · ⊗ ıi−1 ⊗Id⊗ ıi+2 ⊗ · · · ⊗ ıq and ν is the tensor product of the multiplication maps m̄j : Fa1,j ⊗ · · · ⊗Fai−1,j ⊗ Sai,j+ai+1,j F⊗ Fai+2,j ⊗ · · · ⊗ Faq,j → S

λ̃j−µ̃j F, j = 1, . . . , λ1 defined over generators bym̄j( f1 ⊗ · · · ⊗ fi−1 ⊗ fi fi+1 ⊗ fi+2 ⊗ · · · ⊗ fq) = f1 · · · · · fq, where fai,j = 1 ∈ R i f ai,j = 0.Proof. d(λi ,λi+1)/(µi ,µi+1) is the composition of maps ıi ⊗ ıi+1 : Λ

λi−µi F ⊗ Λλi+1−µi+1 F →Fai,1 ⊗ · · · ⊗ Fai,λ1 ⊗ Fai+1,1 ⊗ · · · ⊗ Fai+1,λ1 and the tensor product of the maps m

′j : Fai,j ⊗

Fai+1,j → Sai,j+ai+1,j F, j = 1, . . . , λ1. Then, we can write ν ◦ η ◦ ν = (m̄1 ⊗ · · · ⊗ m̄λ1)(Id⊗m′1 ⊗ · · · ⊗ m′λ1 ⊗ Id)(ı1 ⊗ · · · ⊗ ıq), where (Id ⊗ m

′1 ⊗ · · · ⊗ m′λ1 ⊗ Id) :

⊕ai,j∈αλ/µ Fai,j →⊕

ak,j ,ki+1 Fak,j .Clearly (m̄1 ⊗ · · · ⊗ m̄λ1)(Id⊗m

′1 ⊗ · · · ⊗m′λ1 ⊗ Id) = m. �

As a result of these factorizations we focus our study on dλ/µ when µ = (µ1, µ2) andλ = (λ1, λ2).

Let p1 = λ1 − µ1, p2 = λ2 − µ2 and k = λ2 − µ1, and consider the partitions λ′ =(p1 + p2 − k, p2) and µ′ = (p2 − k). Since λ′/µ′ = (p1, p2) = λ/µ, dλ/µ = dλ′/µ′ .

Definition 2.1.10. Let p1, p2, k ∈N such that pi ≥ k. We define the map

δp1,p2k : Λ

p1 F⊗Λp2 F → S2F⊗ · · · ⊗ S2F⊗Λp1−kF⊗Λp2−kF

as the composition Λp1 F ⊗ Λp2 F ∆⊗∆−−→ ΛkF ⊗ Λp1−kF ⊗ ΛkF ⊗ Λp2−kF Id⊗T⊗Id−−−−−→ ΛkF ⊗

ΛkF ⊗ Λp1−kF ⊗ Λp2−kFd(k,k)⊗Id⊗Id−−−−−−−→ S2F ⊗ · · · ⊗ S2F ⊗ Λp1−kF ⊗ Λp2−kF, where ∆ is the

appropriate diagonal map and T is the canonical isomorphism Λp1−kF ⊗ ΛkF � ΛkF ⊗Λp1−kF.

Lemma 2.1.11. Let p1, p2, k ∈ N such that pi ≥ k + 1. Then δp1,p2k+1 = (Id ⊗ δ

p1−k,p2−k1 ) ◦

δp1,p2k .

Proof. Remember d(k+1,k+1) : Λk+1F⊗Λk+1F∆⊗∆−−→ F⊗ · · · ⊗ F

m1⊗···⊗mk+1−−−−−−−→ S2F⊗ · · · ⊗S2F. By coassociativity and cocommutative of diagonal, ∆⊗ ∆ equals to the map

Λk+1F⊗Λk+1F ∆⊗∆−−→ ΛkF⊗ F⊗ΛkF⊗ F Id⊗T⊗Id−−−−−→ ΛkF⊗ΛkF⊗ F⊗ F ∆⊗∆−−→ F⊗ · · · ⊗F⊗ F⊗ · · · ⊗ F⊗ F⊗ F and then d(k+1,k+1) = d(k,k) ⊗ d(1,1).

We have, δp1,p2k+1 = (m1 ⊗ · · · ⊗mk+1 ⊗ Id⊗ Id)(∆⊗ ∆⊗ Id⊗ Id)(Id⊗ T ⊗ Id)(∆⊗ ∆),where (∆⊗ ∆⊗ Id⊗ Id)(Id⊗ T ⊗ Id)(∆⊗ ∆) : Λp1 F ⊗ Λp2 F → F ⊗ · · · ⊗ F ⊗ F ⊗ · · · ⊗F⊗Λp1−k−1F⊗Λp2−k−1F.


Again by coassociativity and cocommutative, the last map equals to(Id⊗ T ⊗ Id)(∆⊗ ∆⊗ ∆⊗ ∆)(Id⊗ T ⊗ Id)(∆⊗ ∆) : Λp1 F⊗Λp2 F → ΛkF⊗Λp1−kF⊗

ΛkF ⊗ Λp2−kF → ΛkF ⊗ ΛkF ⊗ Λp1−kF ⊗ Λp2−kF → F ⊗ · · · ⊗ F ⊗ F ⊗ · · · ⊗ F ⊗ F ⊗Λp1−k−1F⊗ F⊗Λp2−k−1F → F⊗ · · · ⊗ F⊗ F⊗ · · · ⊗ F⊗ F⊗ F⊗Λp1−k−1F⊗Λp2−k−1F

And then, applying m1 ⊗ · · · ⊗mk ⊗mk+1,δ

p1,p2k+1 = (m1⊗ · · · ⊗mk⊗mk+1⊗ Id)(Id⊗ T⊗ Id)(∆⊗∆⊗∆⊗∆)(Id⊗ T⊗ Id)(∆⊗∆)

which is exactly the map (Id⊗ δp1−k,p2−k1 )δp1,p2k . �

Lemma 2.1.12. Let α = (ai,j) be the 2× p1 + p2 − k matrix of zeros and ones:

1 . . . 1 1 . . . 1 0 . . . 01 . . . 1 0 . . . 0 1 . . . 1︸︷︷︸︸︷︷︸︸︷︷︸

k p1 − k p2 − kThen the diagram

Λp1 F⊗Λp2 F

S2F⊗ · · · ⊗ S2F⊗Λp1−kF⊗Λp2−kF Sα̃F

δp1,p2k

dα

Id⊗∆⊗∆

is commutative and Imδp1,p2k � Imdα. In particular, if λ = (p1 + p2 − k, p2) and µ =(q2 − k, 0), then Imδ

p1,p2k � Lλ/µ.

Proof. First we describe the map dα.

dα : Λp1 F⊗Λp2 Fı1⊗ı2−−−→ ⊕ai,j∈α Fai,j → S2F⊗ · · · ⊗ S2F⊗ F⊗ · · · ⊗ F, since ∑p1+p2−kj=1 a1,j =

p1, ∑p1+p2−kj=1 a2,j = p2 and q1 = 2 = · · · = 2 = qk, qk+1 = 1 = · · · = 1 = qp1+p2−k. More

explicit, we can write dα = (m1 ⊗ · · · ⊗mk ⊗ Id⊗ · · · ⊗ Id)(∆⊗ ∆).Now, (Id⊗ ∆⊗ ∆)δp1,p2k = (Id⊗ · · · ⊗ Id⊗ ∆⊗ ∆)(m1 ⊗ · · · ⊗mk ⊗ Id⊗ Id)(∆⊗ ∆⊗

Id ⊗ Id)(Id ⊗ T ⊗ Id)(∆ ⊗ ∆). It is clear that this map equals to (m1 ⊗ · · · ⊗ mk ⊗ Id ⊗Id)(∆ ⊗ ∆ ⊗ ∆ ⊗ ∆)(Id ⊗ T ⊗ Id)(∆ ⊗ ∆). By coassociativity and commutativity, (Id ⊗· · · ⊗ Id⊗∆⊗∆)(∆⊗∆⊗ Id⊗ Id)(Id⊗ T⊗ Id)(∆⊗∆) : Λp1 F⊗Λp2 F → ΛkF⊗Λp1−kF⊗ΛkF⊗Λp2−kF → ΛkF⊗ΛkF⊗Λp1−kF⊗Λp2−kF → F⊗ · · · ⊗ F equals to the map ∆⊗ ∆ :Λp1 F⊗Λp2 F → ⊕ai,j∈α Fai,j . Directly, (Id⊗ ∆⊗ ∆)δp1,p2k = dα.

Observe that the bottom map Id⊗ ∆⊗ ∆ is an injection, hence Im(δp1,p2k ) � Im(dα). �

Lemma 2.1.13. Let p1, p2, k be positive integers, with pi ≥ k. We define the followingcomposite map:

ωk : Λp1+p2−kF⊗ΛkF ∆⊗Id−−−→ Λp1 F⊗Λp2−kF⊗ΛkF Id⊗m̃−−−→ Λp1 F⊗Λp2 F

where m̃ is the multiplication on the exterior algebra ΛF. Then, the following diagram iscommutative

Λp1+p2−kF⊗ΛkF Λp1 F⊗Λp2 F

S2F⊗Λp1+p2−k−1F⊗Λk−1F S2F⊗Λp1−1F⊗Λp2−1F

δp1+p2−k,k1

wk

δp1,p21

Id⊗wk−1

.


Proof. We can write the map (Id ⊗ ωk−1)δp1+p2−k,k1 as the composition Λ

p1+p2−kF ⊗ΛkF ∆⊗∆−−→ F ⊗ Λp1+p2−k−1F ⊗ F ⊗ Λk−1F Id⊗∆⊗Id⊗Id−−−−−−−→ F ⊗ Λp1−1F ⊗ Λp2−1−(k−1)F ⊗ F ⊗Λk−1F

(Id⊗T⊗T⊗Id)−−−−−−−−→ F ⊗ F ⊗ Λp1−1F ⊗ Λp2−1−(k−1)F ⊗ Λk−1F m⊗Id⊗m̃−−−−−→ S2F ⊗ Λp1−1F ⊗Λp2−1F.

By coassociativity of diagonal, Λp1+p2−kF⊗ΛkF ∆⊗∆−−→ F⊗Λp1+p2−k−1F⊗ F⊗Λk−1FId⊗∆⊗Id⊗Id−−−−−−−→ F ⊗ Λp1−1F ⊗ Λp2−1−(k−1)F ⊗ F ⊗ Λk−1F equals to Λp1+p2−kF ⊗ ΛkF ∆⊗∆−−→

Λp1 F⊗Λp2−1−(k−1)F⊗ F⊗Λk−1F ∆⊗Id⊗Id⊗Id−−−−−−−→ F⊗Λp1−1F⊗Λp2−1−(k−1)F⊗ F⊗Λk−1F.We denote f ⊗ g := f1 ∧ · · · ∧ fp1+p2−k ⊗ gp1+p2−k+1 ∧ · · · ∧ gp1+p2 ∈ Λ

p1+p2−kF⊗ΛkF.Then, (Id ⊗ ωk−1)δ

p1+p2−k,k1 ( f ⊗ g) = ∑i,j,t(−1)sg(i)+sg(j)+sg(t) f ji(1) · gt(1) ⊗ f ji(2) ∧ · · · ∧

f ji(p1) ⊗ fti(p1+1) ∧ · · · ∧ fti(p1+p2−k) ∧ gt(2) ∧ · · · ∧ gt(k).

Similarly, we can write the map δp1,p21 ◦ wk as the composition Λp1+p2−kF⊗ΛkF∆⊗Id−−−→

Λp1 F ⊗ Λp2−kF ⊗ ΛkF Id⊗m̃−−−→ Λp1 F ⊗ Λp2 F ∆⊗∆−−→ F ⊗ Λp1−1F ⊗ F ⊗ Λp2−1F Id⊗T⊗Id−−−−−→ F ⊗F ⊗ Λp1−1F ⊗ Λp2−1F m⊗Id⊗Id−−−−−→ S2F ⊗ Λp1−1F ⊗ Λp2−1F. By the compatibility betweencomultiplication and multiplication on the exterior algebra ΛF the map Λp2−kF⊗ΛkF →Λp2 F → F⊗Λp2−1F equals to

∑1α=0 Λp2−kF⊗ΛkF ∆⊗∆−−→ ΛαF⊗Λp2−k−αF⊗Λ1−αF⊗Λk+α−1F Id⊗T⊗Id−−−−−→ ΛαF⊗Λ1−αF

⊗Λp2−k−αF⊗Λk+α−1F m̃⊗m̃−−−→ F⊗Λp2−1F.Thus, (δp1,p21 ◦wk)( f ⊗ g)=∑i,j,t(−1)sg(i)+sg(j)+sg(t) f ji(1) · fti(p1+1)⊗ f ji(2) ∧ · · · ∧ f ji(p1)⊗

fti(p1+2) ∧ · · · ∧ fti(p1+p2−k) ∧ g + ∑i,j,t(−1)sg(i)+sg(j)+sg(t) f ji(1) · gt(1) ⊗ f ji(2) ∧ · · · ∧ f ji(p1) ⊗

fti(p1+1) ∧ · · · ∧ fti(p1+p2−k) ∧ gt(2) ∧ · · · ∧ gt(k).To finish the proof it is enough to show that the first addend is zero. In fact, the

first term is the image of f ⊗ g under the composite map Λp1+p2−kF⊗ΛkF ∆⊗Id−−−→ Λp1 F⊗Λp2−kF⊗ΛkF ∆⊗∆⊗Id−−−−−→ F⊗Λp1−1F⊗ F⊗Λp2−k−1F⊗ΛkF Id⊗T⊗Id⊗Id−−−−−−−→ F⊗ F⊗Λp1−1F⊗Λp2−k−1F ⊗ ΛkF m⊗⊗Id⊗m̃−−−−−−→ S2F ⊗ Λp1−1F ⊗ Λp2−1. And again because of coassociativ-ity and cocommutativity of comultiplication, Λp1+p2−kF ⊗ ΛkF ∆⊗Id−−−→ Λp1 F ⊗ Λp2−kF ⊗ΛkF ∆⊗∆⊗Id−−−−−→ F⊗Λp1−1F⊗ F⊗Λp2−k−1F⊗ΛkF equals to Λp1+p2−kF⊗ΛkF ∆⊗Id−−−→ Λp1−1F⊗Λp2−k+1F⊗ΛkF Id⊗∆⊗Id−−−−−→ Λp1−1F⊗Λ2F⊗Λp2−k−1F⊗ΛkF Id⊗∆⊗Id⊗Id−−−−−−−→ Λp1−1F⊗ F⊗F⊗Λp2−k−1F⊗ΛkF

A simple calculation shows that the image which we are looking for is ∑i,j fi(1) ∧ · · · ∧fi(p1−1) ⊗ ( f ji(p1) · f ji(p1+1) − f ji(p1+1) · f ji(p1)))⊗ f ji(p1+2) ∧ · · · ∧ f ji(p1+p2−k), which is zerosince u · v = v · u ∈ S2F. �

Proposition 2.1.14. Let p1, p2, k ∈N with pi ≥ k + 1. Then the composition

Λp1+p2−kF⊗ΛkF wk−→ Λp1 F⊗Λp2 Fδ

p1,p2k+1−−−→ S2F⊗ · · · ⊗ S2F⊗Λp1−k−1F⊗Λp2−k−1

is zero.Proof. We proceed by induction on k. When k = 0, we have Λp1+p2 ∆−→ Λp1 ⊗Λp2 ∆⊗∆−−→F ⊗ Λp1−1F ⊗ F ⊗ Λp2−1F → S2F ⊗ Λp1−1F ⊗ Λp2−1F. The same argument we have justseen in the last part of the proof of Lemma 2.1.13 shows that this composition is zero.Suppose k > 0. Since δp1,p2k+1 = (Id⊗ δ

p1−k,p2−k1 ) ◦ δ

p1,p2k , we can write δ

p1,p2k+1 ◦ wk = (Id⊗


δp1−k,p2−k1 ) ◦ δ

p1,p2k ◦ wk. By Lemma 2.1.11 (Id ⊗ δ

p1−k,p2−k1 ) ◦ δ

p1,p2k = (Id ⊗ δ

p1−k,p2−k1 ) ◦

(Id ⊗ wk−1) ◦ δp1+p2−k,k1 . The result follows using the induction on (Id ⊗ δ

p1−k,p2−k1 ) ◦

(Id⊗ wk−1). �

Lemma 2.1.15. Let p1, p2, k, u, l be nonnegative integers such that pi ≥ k, 0 ≤ u ≤ l ≤k− 1. Denote by ω̄u the composite map

ΛuF⊗Λp1+p2−l F⊗Λl−uF Id⊗∆⊗Id−−−−−→ ΛuF⊗Λp1−uF⊗Λp2+u−l F⊗Λl−uF m̃⊗m̃−−−→ Λp1 F⊗Λp2 F.

Then Im(ω̄u) is contained in the image of the map

ωp1,p2k :

⊕0≤v≤k−1

Λp1+p2−vF⊗ΛvF∑k−1v=0 ωv−−−−→ Λp1 F⊗Λp2 F.

Proof. We proceed by induction on u. When u = 0, w̄u = wl and the propositionis clear. Assuming u > 0, let x ⊗ y ⊗ z ∈ ΛuF ⊗ Λp1+p2−l F ⊗ Λl−uF, we have ω̄u(x ⊗y ⊗ z) = ∑i(−1)sg(i)x1 ∧ · · · ∧ xu ∧ yi(1) ∧ · · · ∧ yi(p1−u) ⊗ yi(p1−u+1) ∧ · · · ∧ yi(p1+p2−l) ∧z1 ∧ · · · ∧ zl−u. Now, we consider the composition ωl−u ◦ (m⊗ Id) : ΛuF ⊗ Λp1+p2−l F ⊗Λl−u m⊗Id−−−→ Λp1+p2−(l−u)F⊗Λl−uF ∆⊗Id−−−→ Λp1 F⊗Λp2−(l−u)F⊗Λl−uF Id⊗m−−−→ Λp1 F⊗Λp2 F.Because of compatibility between the multiplication and comultiplication the composition

ΛuF⊗Λp1+p2−l F m−→ Λp1+p2−(l−u)F ∆−→ Λp1 F⊗Λp2−(l−u)F equals to∑uα=0 Λ

uF⊗Λp1+p2−l F ∆⊗∆−−→ Λu−αF⊗ΛαF⊗Λp1−(u−α)F⊗Λp2−(l−u+α)F Id⊗T⊗Id−−−−−→ Λu−αF⊗Λp1−(u−α)F⊗ΛαF⊗Λp2−(l−u+α)F m⊗m−−−→ Λp1 F⊗Λp2−(l−u)F.

Thus ωl−u ◦ (m⊗ Id)(x⊗ y⊗ z) = ω̄u(x⊗ y⊗ z) + ∑uα=1 ∑i,j(−1)sg(i)+sg(j)xi(1) ∧ · · · ∧xi(u−α) ∧ yj(p1) ∧ · · · ∧ yj(1) ∧ · · · ∧ yj(p1−(u−α)) ⊗ xi(u−α+1) ∧ · · · ∧ xi(u) ∧ yj(p1−(u−α)+1 ∧· · · ∧ yj(p2+p1−l). Since ωα(x ⊗ y ⊗ z) = ∑i(−1)

sg(i)xi(1) ∧ · · · ∧ xi(u−α) ∧ y ∧ xi(u−α+1) ∧· · · ∧ xi(u) ∧ z, we obtain ∑uα=1 ω̄u−α ◦ωα(x⊗ y⊗ z) = ωl−u ◦ (m⊗ Id)(x⊗ y⊗ z)− ω̄u(x⊗y⊗ z). By induction Im(ω̄u−α ◦ ωα) ⊂ Im(ωp1,p2k ), ∀1 ≤ α ≤ u, and hence ωl−u ◦ (m⊗Id)(x⊗ y⊗ z) ∈ Im(ωp1,p2k ). �

At this moment we define the auxiliary map we have mentioned at the begging of thissubsection and next we state our first main result.

Definition 2.1.16. Let λ = (λ1, . . . , λq) and µ = (µ1, . . . , µq) be two partitions such thatµ ⊆ λ. For each i = 1, . . . , q− 1 consider partitions λi = (λi, λi+1) and µi = (µi, µi+1). Onedefines a map ωλ/µ to be the sum of maps Id1 ⊗ · · · ⊗ Idi−1 ⊗ ωλi/µi ⊗ Idi+2 ⊗ . . .⊗ Idq,where i = 1, . . . , q− 1 and ωλi/µi = ω

pi ,pi+1ki

with pi = λi − µi, ki = λi+1 − µi. We defineL̄λ/µ(F) to be the cokernel of map ωλ/µ.

Theorem 2.1.17. The image of ωλ/µ is contained in the kernel of dλ/µ.Proof. Keeping in mind the factorization of dλ/µ, Lemma 2.1.9 it is enough to considerλ = (λ1, λ2), µ = (µ1, µ2) and show that Im(ω

p1,p2k ) ⊆ ker(dλ/µ).

Now, since the composition δp1,p2k+1 ◦ ωk = 0 as we have seen before, ωk ⊂ ker(δp1,p2k+1 )

and hence Im(∑k−1v=0 ωv) ⊆ ker(∑k−1v=0 δ

p1,p2v+1 ). We have already finished considering that for

each k ∈ N | pi ≥ k, (Id⊗ ∆⊗ ∆) ◦ δp1,p2k = dλ/µ ⇒ ker(δ

p1,p2k ) ⊆ ker(dλ/µ) and then

Im(∑k−1v=0 ωv) ⊂ ker(∑k−1v=0 δ

p1,p2v+1 ) ⊂ ker(dλ/µ). �


As a direct corollary of Theorem 2.1.17 the map dλ/µ induces a surjective map d̄λ/µ :L̄λ/µ(F)→ Lλ/µ(F) defined by d̄λ/µ(x̄) = dλ/µ(x), where x̄ is the image under the naturalmap π′ : Λλ/µF → L̄λ/µF. Note that x̄ = ȳ ⇒ d̄λ/µ(x̄) = d̄λ/µ(ȳ) ⇔ dλ/µ(x) = dλ/µ(y)which implies dλ/µ = d̄λ/µ ◦ π′.

Lemma 2.1.18. Let T ∈ Tabλ/µ(S) be a row-standard tableau which is not standard. Thenthere exist standard tableaux Ti with Ti < T such that XT −∑±XTi ∈ Im(wλ/µ).Proof. Let T j = (T(j, µj + 1), . . . , T(j, λj)) and T j+1=(T(j+ 1, µj+1 + 1), . . . , T(j+ 1, λj+1)be the first two rows of T in which column-standardness is violated. Let T̄ = (T j, T j+1)be a tableau of shape λj/µj where λj = (λj, λj+1), µj = (µj, µj+1). First, we will proof thelemma in this particular case.

Let p1 = λj − µj, p2 = λj+1− µj+1 and k = λj+1− µj. For convenience, we will denoteT(j, i) = ai, i = 1, . . . , p1 and T(j + 1, l) = bl , l = 1, . . . , p2. Let au+1 > bp2−k+u+1be the first entries in which the violation takes place. Let X̄ = a1 ∧ · · · ∧ au ⊗ b1 ∧ · · · ∧bp2−k+u+1 ∧ au+1 ∧ · · · ∧ ap1 ⊗ bp2−k+u+2 ∧ · · · ∧ bp2 ∈ Λ

uF ⊗ Λp1+p2−l F ⊗ Λl−1−uF, withl = k− 1. Graphically:

a1 . . . au+1 . . .b1 . . . bp2−k bp2−k+1 . . . bp2−k+u+1 . . .

We have w̄u(X̄) = ∑(−1)sg(i)a1 ∧ · · · ∧ au ∧ cI ⊗ cI′ ∧ bp2−k+u+2 ∧ · · · ∧ bp2 =: ∑±X̄Iwhere cI is the corresponding exterior product of p1 − u terms of b1 ∧ · · · ∧ bp2−k+u+1 ∧au+1 ∧ · · · ∧ ap1 and cI′ of the complementary terms after applying Id⊗ ∆⊗ ∆. We denotecI = ci1 ∧ · · · ∧ cip1−u and cI′ = cip1−u+1 ∧ · · · ∧ cp1+p2−k+u+1. By Lemma 2.1.15, ∑±X̄I ∈Im(wp1,p2k ) = Im(ωλj/µj).

For each I let T̄I be the tableau of shape λj/µj associated to X̄I . There is only oneterm I, say I0, such that cI = au+1 ∧ · · · ∧ ap1 , and hence the tableau associated to I0is the tableau associated to X̄. The rest of tableaux T̄I are a result of exchanging someentries au+1, . . . , ap1 by some entries b1, . . . , bp2−k+u+1, with necessarily ci1 = bt for somet ∈ {1, . . . , p2 − k + u + 1}, by a previous lemma T̄I < T̄.

Observe that each a1 ∧ · · · ∧ au ∧ cI ⊗ cI′ ∧ bp2−k+u+2 ∧ · · · ∧ bp2 as an element of Λp1 F⊗

Λp2 F can be arranged such that equals xI1 ∧ · · · ∧ xIp1 ⊗ xI′1 ∧ · · · ∧ xI′p2 with xI1 < . . . < xIp1and xI′1 < . . . < xI′p2 . Because of this we assume that T̄I , I , I0 is row-standard. In thissituation for each T̄I we have ci1 ≤ cp2−k+u+1, since au+1 > bp2−k+u+1 ⇒ au+1 > bt, t =1, . . . , p2 − k + u + 1. So, if T̄I is not standard, then the first index where the columnstandardness violation in T̄I takes place is greater than u + 1.

Since there is a finite number of tableaux in Tabλj/µj(S), repeating the same procedurefor each non standard tableau T̄I and continue as many times we need, finally we willobtain ∑ X̄TI = X̄−∑I,I0 ±X̄TI ∈ Im(ωλj/µj).

Now the general case is clear, we only have to exchange X̄ by T(1, µ1 + 1) ∧ · · · ∧T(1, λ1)⊗ · · · ⊗ X̄ ⊗ T(j + 2, µj+2 + 1) ∧ · · · ∧ T(j + 2, λj+2)⊗ · · · , w̄u by the natural gen-eralization Id1 ⊗ · · · ⊗ Idi−1 ⊗ w̄ui ⊗ Idi+2 ⊗ · · · ⊗ Idq, wλj/µj by wλ/µ and repeat this pro-cedure for each couple of rows which column-standardness is violated. �

Let us to apply the above lemma to the element associated to the tableau T = x1 x6x2 x5

.


We have, p1 = 2, p2 = 2, k = 2, u = 1 and l = 1. We want to compute ω̄2(x1⊗ x2 ∧ x5 ∧x6), where ω̄2 is the map F⊗Λ3F

Id⊗∆−−−→ F⊗ F⊗Λ2F m⊗Id−−−→ Λ2F⊗Λ2F. We obtain,

x1 x2x5 x6

− x1 x5x2 x6

+ x1 x6x2 x5

Directly from Lemma 2.1.18

Corollary 2.1.19. {dλ/µ(XT) | T ∈ Tabλ/µ(S) is standard} is a system of generators inLλ/µF.

Corollary 2.1.20. {X̄T | T ∈ Tabλ/µ(S) is standard} spans L̄λ/µF.Proof. Since {XT | T ∈ Tabλ/µ(S) is row-standard} generate Λλ/µ(F), it is enough tosee that each X̄T , with T row-standard can be expressed as a combination of elements ofthat such.

Let T ∈ Tabλ/µ(S) be a row-standard tableau. By Lemma 2.1.18 we know that thereexist TI ∈ Tabλ/µ(S) standard tableaux such that XT −∑I XTI ∈ Im(ωλ/µ). Clearly, X̄T =∑I X̄TI . �

Theorem 2.1.21. {dλ/µ(XT) | T ∈ Tabλ/µ{x1, . . . , xn} is standard} is a basis of Lλ/µFcalled the standard basis of Lλ/µF. So, Lλ/µF is a free R-module.Proof. We denote λ = (λ1, . . . , λq), µ = (µ1, . . . , µq). Since {dλ/µ(XT) | T ∈ Tabλ/µ{x1,. . . , xn} is standard} is a system of generators of Lλ/µ as we have just seen, we only mustproof that these elements are independent. We proceed as follows:

Let Λ′λ/µF and S′λ̃/µ̃

F be the free R-submodules of Λλ/µ and Sλ̃/µ̃F respectively gen-

erated by {XT | T ∈ Tabλ/µ(S) is standard} and {ZT | T ∈ Tabλ/µ(S) respec-tively. Consider the natural injection i : Λ′λ/µF ↪→ Λλ/µF and the natural projectionπ : Sλ̃/µ̃F → S′λ̃/µ̃F. Keeping in mind that Tabλ/µ(S) is totally ordered lexicographi-cally and dλ/µ(Λ′λ/µF) = Lλ/µF , if we prove that the map associated to the compositionπ ◦ dλ/µ ◦ i is triangular with ones on the diagonal respect these basis in this order, thendλ/µ ◦ i is an injection and hence Lλ/µF � L′λ/µF which implies that {dλ/µ(XT) | T ∈Tabλ/µ{x1, . . . , xn} is standard} is a basis of Lλ/µF.

Let T ∈ Tabλ/µ(S) be a standard tableau. As we have seen at the beginning ofthis section, dλ/µ(XT) = ∑σ(−1)sg(σ)ZTσ where σ = (σ1, . . . , σq), σi runs through allpermutations of {µi + 1, . . . , λi} and Tσ ∈ Tabλ/µ{x1, . . . , xn} is the tableau of entriesTσ(i, j) = T(i, σi(j)). Let T′σ be the column standardization of Tσ, since T′σ and Tσ definethe same element ZT in Sλ̃/µ̃F because the properties of the symmetric power, we canwrite dλ/µ(XT) = ∑σ ZTσ′ where σ

′ runs over the set of permutations such that Tσ′ iscolumn-standard. Then π ◦ dλ/µ(XT) = ∑σ ZT′σ which σ runs over the set of permutationssuch that T′σ is standard.

We observe that ZT occurs in the sum above, then it is enough to prove that T′σ < T forall σ , Id. First we observe that Tσ cannot be row-standard since the standardness of T,and hence Tσ , T′σ if σ , Id. For σ , Id, T′σ is obtained from Tσ as a result of iterating theprocess described in Lemma 2.1.8, so T′σ < Tσ, and by the same reason Tσ < T. Clearlythis implies T′σ < T for all σ , Id. �

The above theorem give us an explicit basis of Schur functors by means of combinatorictools. As a particular case we have the following theorem (see [15], Theorem 6.3):

2.2 CoSchur Functors and the freeness of the CoSchur functors. 19

Theorem 2.1.22. Let λ = (λ1, . . . , λq) be a partition. If λ1 ≤ n, then

dim(LλF) = ∏1≤i


Definition 2.2.1. Let λ = (λ1, . . . , λp) and µ = (λ1, . . . , λp) be two partitions with µ ⊆ λ.We define the coSchur map d′λ/µ : Dλ/µF → Λλ/µF associated to the skew partition λ/µ to

be the composite map Dλ1−µ1 F ⊗ · · · ⊗ Dλp−µp F α′−→ ⊗(i,j)∈∆λ/µ F(i,j) β′−→ Λλ̃1−µ̃1 F ⊗ · · · ⊗

Λλ̃λ1−µ̃λ1 , where α′ is the tensor product of the inclusions ı′i : Dλi−µi F → F(i,µi+1) ⊗ · · · ⊗

F(i,λi), i = 1, . . . , p and β is the tensor product of the multiplications m̃j : F(µ̃1+1,j) ⊗ · · · ⊗F(λ̃i ,j) → Λ

λ̃j−µ̃j F, j = 1, . . . , λ1.

Definition 2.2.2. We define the coSchur functor Kλ/µF associated to F and the skew parti-tion λ/µ to be the image of d′λ/µ.

Clearly, Kλ/µF can be defined through Ferrers matrices in analogous way as Lλ/µ andthe coSchur map can be generalized as a natural map d′α where α is an arbitrary matrix ofzeros as we have seen in previous sections.

Examples 2.2.3. (1) K(t)F = DtF.

(2) K(1,...,1)F = Λ(t)F, t = 1 + · · ·+ 1.

The Standard Basis Theorem of Schur functors has its dual version to CoSchur functors.Indeed, we easily check that {d′λ/µ(XT) | T ∈ Tabλ/µ is co-column-standard} is a basisof Dλ/µ and arguing as in section 2.1.1 we prove:

Theorem 2.2.4. {d′λ/µ(XT) | T ∈ Tabλ/µ{x1, . . . , xn} is co-standard} is a basis of Kλ/µFcalled the co-standard basis of Kλ/µF. So, Kλ/µF is a free R-module.

The functoriality of Kλ/µ(−) follows from the functoriality of Dr(−) in the same man-ner we have argued from Schur functors in section 2.1.1. We will finish our generaldiscussion of Schur and CoSchur functors by stating the duality (Lλ/µF)∗ � Kλ̃/µ̃(F

∗). Itfollows directly from the dual nature of the Schur and CoSchur maps.

Theorem 2.2.5. (Lλ/µF)∗ � Kλ̃/µ̃(F∗).

Proof. We denote d′λ̃/λ̃

: Dλ̃/µ̃F → Λλ/µF and dλ/µ : Λλ/µF → Sλ̃/µ̃F . The coSchur mapis the composite map β′ ◦ α′, where β′ is the tensor product of diagonal maps in DF andα′ the tensor product of multiplications in ΛF. Similarly we have dλ/µ = α ◦ β. From theIntroductory Material, β′ is the dual map of α and α′ is the dual map β. �

Moreover, if R contains a field of characteristic zero, then Lλ/µF � Kλ̃/µ̃F. See [2],Corollary (2.3.3). From Theorem 2.2.5, it follows that Kλ/µF � (Lλ̃/µ̃F

∗)∗. This isomor-phism give the following formula of rank of CoSchur functor.

Theorem 2.2.6. Let λ = (λ1, . . . , λq) be a partition. If λ̃1 ≤ n, then

dim(KλF) = ∏1≤i

2.3 Decomposition of Schur Functors. 21

2.3 Decomposition of Schur Functors.

The aim of this section will be to define a filtration on Lλ/µ(F ⊕ G) with associatedgraded module

⊕µ⊆γ⊆λ Lγ/µF⊗ Lλ/γG.

Let F and G be free R-modules of rank n and m with ordered basis {x1, . . . , xn} and{xn+1, . . . , xn+m} respectively. We will consider the ordered basis {x1, . . . , xn+m} in F⊕G.

Definition 2.3.1. Let F be a module over a ring R. A finite filtration of F is a sequence ofsubmodules of F

0 =: F0 ⊂ F1 ⊂ · · · ⊂ Fn−1 ⊂ Fn := F.

One defines the associated graded object of the filtration ∑n−1i=0 Fi+1/Fi.

Once we have defined the modules of the filtration, we will need to describe basis ofthese modules. We do it through a basis of F⊕G and Tabλ/µ(F⊕G). We start introducingsome facts about partitions and tableaux we will use in this description.

Definition 2.3.2. Let λ = (λ1, . . . , λq), µ = (µ1, . . . , µq) be two partitions. We say λ ≥ µ ifλ1 = µ1, . . . , λi = µi and λi+1 > µi+1 for some i.

For example, (1, 1, 1, 1, 3, 5) ≥ (1, 1, 1, 1, 2, 5).

Proposition 2.3.3. ≥ is a total order on N∞.Proof. First we will see that ≥ is an order. Clearly, λ ≥ λ for all partition λ ∈ N∞. Letλ, µ be two partitions such that λ ≥ µ and µ ≥ λ and let i, j be the indexes such thatλ1 = µ1, . . . , λi = µi, λi+1 > µi+1 and µ1 = λ1, . . . , λj = µj, µj+1 > λj+1. Clearly wemust have i = j = q. Finally, let λ, µ and σ be partitions such that λ ≥ µ and µ ≥ σ.As before, we say λi > µi and µj > σj. Then, λ1 = µ1, . . . , λi−1 = µi−1, λi > µi, andµ1 = σ1, . . . , µj−1 = σj−1, µj > σj. It is enough to take k = min{i, j}.

And second, we will show that ≥ is a total order. Given λ, µ two partition we sayi = max{i | λ1 = µ1, . . . , λi = µi}. We will see that λ ≤ µ or µ ≤ λ. The case i = q isclearly, so we suppose i < q. Since we have λi+1 ≥ µi+1 or µi+1 ≥ λi+1, the result follows.

�

Definition 2.3.4. Let E, F be two free modules with basis{x1, . . . , xm}and{xm+1, . . . , xm+n}respectively and let T ∈ Tabλ/µ({x1, . . . , xm+n}). For each i ∈ {1, . . . , q} let ηi to be µiplus the number of basis elements of F in the ith row of T. We define the sequenceη(T) = (η1, . . . , ηq) ∈N∞.

Observe that when T ∈ Tabλ/µ(S) is standard, then η(T) is a partition and µ ⊆ η(T) ⊆λ. Indeed, let Ti, Ti+1 be two rows of T. We say ηi(T) = µi + ki where ki is the numberof basis elements of F in the ith row. If T(i, l) is a basis element of E, then T(i + 1, l)must be a basis element of E too since T is column-standard and thus T(i, l) ≤ T(i + 1, l)(remember the ordered basis {x1, . . . , xn, xn+1, . . . , xn+m}). Necessarily, ki ≥ ki+1 and thenµi + ki ≥ µi+1 + ki+1.

Lemma 2.3.5. Let S, T ∈ Tabλ/µ{x1, . . . , xn+m} with S ≤ T. Then η(S) ≥ η(T).


Proof. We assume η(S) , η(T), otherwise there is nothing to prove. Let k the first integersuch that ηk(S) , ηk(T), we will see ηk(S) > ηk(T). Remember that S ≤ T if Sp,q ≥ Tp,q forall p, q, where Sp,q and Tp,q are the number of times the first q elements of the basis appearin the first p rows of S and T respectively. From the definition of η(S) and η(T) it is clearthat Sp,m = ∑

pj=1 ηj(S)− µj and Tp,m = ∑

pj=1 ηj(T)− µj, and hence ηi(S) = ηi(T), ∀i < k

implies Sp,m = Tp,m, ∀p < k.Writing Sk,m = ∑k−1j=1 ηj(S)− µj + ηk(S)− µk and Tk,m = ∑

k−1j=1 ηj(T)− µj + ηk(T)− µk,

since we have assumed that ηi(S) = ηi(T), i = 1 . . . , k− 1, then Sk,m > Tk,m ⇒ ηk(S) >ηk(T). �

Proposition 2.3.6. Let T ∈ Tabλ/µ(F⊕ G). Then there exist unique standard tableaux Ti,and unique integers ci , 0, such that dλ/µ(XT) = ∑ cidλ/µ(XTi ) and η(Ti) ≥ η(t).Proof. Directly from Lemma 2.3.5 and the standard basis theorem Theorem 2.1.21. �

Definition 2.3.7. Let µ, γ and λ be partitions such that µ ⊆ γ and γ ⊆ λ. We definesubmodules

Mγ(Λλ/µ(F⊕ G)) := Im(ϕ :⊕

µ⊆σ⊆λ,σ≥γΛσ/µ(F)⊗Λλ/σ(G)→ Λλ/µ(F⊕ G)),

Ṁγ(Λλ/µ(F⊕ G)) := Im(ϕ′ :⊕

µ⊆σ⊆λ,σ>γΛσ/µ(F)⊗Λλ/σ(G)→ Λλ/µ(F⊕ G)).

where ϕ and ϕ′ are the sum of maps obtained by tensoring the maps Λσi−µi F⊗Λλi−σi G →Λλi−µi (F⊕G) defined by x1 ∧ · · · ∧ xσi−µi ⊗ y1 ∧ · · · ∧ yλi−σi → x1 ∧ · · · ∧ xσi−µi ∧ y1 ∧ · · · ∧yλi−σi . We define submodules:

Mγ(Lλ/µ(F⊕ G)) := dλ/µ(Mγ(Λλ/µ(F⊕ G))),

Ṁγ(Lλ/µ(F⊕ G)) := dλ/µ(Ṁγ(Λλ/µ(F⊕ G))).

The modules Mγ(Lλ/µ(F⊕ G)) are the submodules of the filtration and the quotientsMγ(Lλ/µ(F⊕G))/Ṁγ(Lλ/µ(F⊕G)) define the graded object. Clearly {XT′ ⊗XT′′ | T′ ∈Tabσ/µ{x1, . . . , xn}, T′′ ∈ Tabλ/σ{xn+1, . . . , xn+m} are row- standard, σ ≥ γ} and {XT′ ⊗XT′′ | T′ ∈ Tabσ/µ{x1, . . . , xn}, T′′ ∈ Tabλ/σ{xn+1, . . . , xn+m} are row-standard, σ > γ}represent basis of the modules

⊕µ⊆σ≥λ,σ≥γ Λσ/µ(F)⊗Λλ/σ(G) and

⊕µ⊆σ≥λ,σ>γ Λσ/µ(F)

⊗Λλ/σ(G) respectively.Note that, ϕ(XT′ ⊗ XT′′) = XT′′ where T′′ ∈ Tabλ/µ(S′′) such that T′′(i, j) = T(i, j) i f

j ≤ σi and T′′(i, j + σi) = T′(i, j) i f j ≤ λi which clearly is a row-standard tableauwith η(T′′) = σ and then η(T′′) ≥ γ or η(T′′) > γ depending on the case. Now, if T ∈Tabλ/µ({x1, . . . , xn+m}) is a standard tableau such that η(T) ≥ γ and hence µ ⊆ η(T′′) ⊆λ, T′(i, j) = T(i, j) i f j ≤ η(T)i defines a standard tableau of shape σ/µ and T′′(i, j) =T(i, j + η(T′′)i) i f j ≤ λi defines a standard tableau of shape λ/σ. Obviously, XT′′ is theimage of XT ⊗ XT′ by ϕ. It follows that {dλ/µ(XT) | T ∈ Tabλ/µ is standard, η(T) ≥ γ}form an R-basis of Mγ(Lλ/µ(F⊕G)) and {dλ/µ(XT) | T ∈ Tabλ/µ is standard, η(T) > γ}form an R-basis of Mγ(Lλ/µ(F⊕ G)).

2.3 Decomposition of Schur Functors. 23

Proposition 2.3.8. The map Λγ/µ(F)⊗Λλ/γ(G)ψ−→ Mγ(Λλ/µ(F⊕ G)) induces a map

ψ̃γ : Lγ/µ(F)⊗ Lλ/γ(G)→ Mγ(Lλ/µ(F⊕ G))/Ṁγ(Lλ/µ(F⊕ G)).

Proof. The map ψ̃γ is defined by sending dγ/µ(x)⊗ dλ/γ(y) to the class of dλ/µ(ψ(x⊗ y)).We only need to verify that ψ̃γ sends 0 to 0̄, since ker(dλ/µ) = Im(wλ/µ), it is sufficientto prove that ψ(Im(wγ/µ)⊗ Λλ/γG) and ψ(Λγ/µ(F)⊗ Im(wλ/γ)) are both contained inN := Im(wλ/µ) + Ṁγ(Lλ/µ(F⊕ G)).

Remember wγ/µ = ∑q−1i=1 Id1 ⊗ . . .⊗ Idi−1 ⊗ wγi/µi ⊗ Idi+2 ⊗ . . .⊗ Idq where wγi/µi =

∑γi+1−µi−1t=0 Λ

γi−µi+γi+1−µi+1−tF ⊗ ΛtF → Λγi−µi F ⊗ Λγi+1−µi+1 F. By the change l = γi −µi+1 − t, we can write the last map as wγi/µi = ∑

γi+1−µi+1l=µi−µi+1+1 Λ

γi−µi+l F⊗Λγi+1−µi+1−l F →Λγi−µi F⊗Λγi+1−µi+1 F. And thus, wγ/µ = ∑

q−1i=1 ∑

γi+1−µi+1l=µi−µi+1+1 Λ

γ1−µ1 F⊗ · · · ⊗Λγi−µi+l F⊗Λγi+1−µi+1−l F ⊗ · · · ⊗ Λγq−µq F → Λγ/µF. Fixing i and l, if we see that ψ(Im(Λγ1−µ1 F ⊗· · ·⊗Λγi−µi+l F⊗Λγi+1−µi+1−l F⊗ · · ·⊗Λγq−µq F⊗Λλ/γG) ⊆ N, clearly we will have showthat ψ(Im(wγ/µ)⊗Λλ/γG) ⊆ N.

Let X = xI1 ⊗ · · · ⊗ xIi−1 ⊗ xIi ⊗ xIi+1 ⊗ xIi+2 ⊗ · · · ⊗ XIq and Y = yJ1 ⊗ · · · ⊗ yJq bebasis elements of Λγ1−µ1 F⊗ · · · ⊗Λγi−µi+l F⊗Λγi+1−µi+1−l F⊗ · · · ⊗Λγq−µq F and Λλ/γGrespectively. Using the coassociativity and commutativity of the comultiplication map, wecan write wl(xIi ⊗ xIi+1) = ∑U ±xU ⊗ xIi+1 ∧ xU′ , where U runs over all subsets of orderγi − µi of Ii and U′ is the complement of U in Ii. Keeping this in mind,

ψ(X ⊗ Y) = ∑U ±xI1 ∧ yI1 ⊗ · · · ⊗ xU ∧ yJi ⊗ xIi+1 ∧ xU′ ∧ yJi+1 ⊗ · · · ⊗ xIq ∧ yIq =:∑U ±ZTUwhere each TU ∈ Tabλ/µ{x1, . . . , xn+m} and clearly η(TU) = γ.

Applying the same decomposition to ωλ/µ and fixing the same i and l, if we considerthe basis element W := xI1 ∧ yI1 ⊗ · · · ⊗ xIq ∧ yIq in Λλ1−µ1(F ⊕ G)⊗ · · · ⊗ Λλi−µi+l(F ⊕G)⊗Λλi+1−µi+1−l(F⊕ G)⊗ · · · ⊗Λλq−µq(F⊕ G) we obtain that the corresponding factorof wλ/µ send that basis element to ∑w1,w2 ±xI1 ∧ yI1 ⊗ · · · ⊗ xw1 ∧ yw2 ⊗ xw′1 ∧ xIi+1 ∧ yw′2 ∧yJi+1 ⊗ · · · ⊗ xIq ∧ yIq where w1, w2 are subsets of Ii and Ji whose orders add up to λi − µiand w′1, w

′2 are the complements of w1, w2 in Ii, Ii+1 respectively.

Since Ji is of order λi − γi, then w1 must be of order ≥ λi − µi − λi + γi = γi − µi, thenwe can write the image above as:

∑U±XTU + ∑

w1,w2±xI1 ∧ yI1 ⊗ · · · ⊗ xw1 ∧ yw2 ⊗ xw′1 ∧ xIi+1 ∧ yw′2 ∧ yJi+1 ⊗ · · · ⊗ xIq ∧ yIq

where now w1 + w2 > λi − µi. Each summand of the second addend corresponds to anelement WTw1,w2 where Tw1,w2 is a tableau satisfying η(Tw1,w2) > γ, since w1 > γi − µi.Then, the second summand is contained in Ṁ(Λλ/µ(F⊕G)) and ψ(X⊗Y) = ωλ/µ(W)−∑w1,w2 WTw1,w2 which clearly is an element of N.

The proof that ψ(Λγ/µF ⊗ Im(wλ/γ)) ⊆ N proceeds formally in the same way thatψ(Im(wγ/µ) ⊗ Λλ/γG) ⊆ N. However we need to define appropriate maps instead ofωλ/γ and ωλ/µ. Observe that considering the maps ωλi/γi = ∑

λi+1−γi+1l=γi−γi+1+1 Λ

λi−γi+l F ⊗

Λλi+1−γi+1−l F → Λλi−γi F ⊗ Λλi+1−γi+1 F and wλi/µi = ∑λi+1−µi+1l=µi−µi+1+1 Λ

λi−µi+l(F ⊕ G) ⊗


Λλi+1−µi+1−l(F⊕G)→ Λλi−µi (F⊕G)⊗Λλi+1−µi+1(F⊕G), there can be some l > γi−γi+1which not occurs at wλi/µi .

Let λi − λi+1 + 1 ≤ t ≤ λi − γi and u := λi − γi − t =: l ≤ λi+1− γi − 1. We define theω̃λ/γ = ∑t w̃u,t where w̃u,t : Λλi−γi−tG⊗Λλi+1−γi+1+tG → Λλi−γi G⊗Λλi+1−γi+1 G whoseimage is contained in the image of the map wλi/γi , for all t. In the same way we defineω̃λ/µ considering µi − µi+1 + 1 ≤ t ≤ λi − µi, u = λi − µi − t = t ≤ λi+1 − µi − 1. Theremainder of the proof is exactly the same as in the first part but replacing ωλ/γ and ωλ/µby ω̃λ/γ and ω̃λ/µ respectively. �

Theorem 2.3.9. The map ψ̃γ is an isomorphism.Proof. ψ̃γ is clearly surjective since {dλ/µ(XT) | T is standard, η(T) = γ} representan R-basis of Mγ(Lλ/µ(F ⊕ G))/Ṁγ(Lλ/µ(F ⊕ G)). Indeed, if T is a standard basis ofTabλ/µ{x1, . . . , xn+m} with η(T) = γ, then T′(i, j) = T(i, j) i f j ∈ {µi + 1, . . . , γi} andT′′(i, j) = T(i, j) i f j ∈ {γi + 1, . . . , λi} defines standard tableaux of Tabγ/µ{x1, . . . , xn}and Tabλ/γ{xn+1, . . . , xn+m} respectively. Then, ψ̃γ(dγ/µ(XT′)⊗ dλ/γ(XT′′))=dλ/µ(ψ(XT′⊗XT′′)) = dλ/µ(XT).

Moreover, ψ̃ carries a basis element of Lγ/µF⊗ Lλ/µG to a basis element of Mγ(Lλ/µ(F⊕G))/Ṁγ(Lλ/µ(F⊕G)). If T′ ∈ Tabγ/µ{x1, . . . , xn} and T′′ ∈ Tabλ/γ{xn+1, . . . , xn+m} arestandard tableaux, ψ(XT′ ⊗ XT′′) = XT where T a standard tableau with η(T) = γ, as wehave seen before. �

Corollary 2.3.10. The submodules {Mγ(Lλ/µ(F⊕ G)) | µ ⊆ γ ⊆ λ} give a filtration ofLλ/µ(F⊕ G), whose associated graded module is isomorphic to

⊕µ⊆γ⊆λ Lγ/µF⊗ Lλ/γG.

Proof. Clearly, {Mγ(Lλ/µ(F ⊕ G)) | µ ⊆ γ ⊆ λ} is a filtration of Lλ/µ(F ⊕ G) withthe total order on partitions which we defined at the beginning of this section . IfMγ(Lλ/µ(F ⊕ G)) ⊂ Mγ′(Lλ/µ(F ⊕ G)) is a piece of this filtration, then γ = max{σ ∈N | γ < γ′}, and hence Mγ(Lλ/µ(F⊕ G)) equals to Ṁγ(Lλ/µ(F⊕ G)). �

Let us see a couple of examples.

Examples 2.3.11. (1) We first start computing the decomposition of L(4,2)/(2,1)(F⊕ G).

From the above discussion we only need to determinate all the partitions γ such thatµ ⊆ γ ⊆ λ. These partitions are (2, 1), (2, 2), (3, 1), (3, 2), (4, 1) and (4, 2). Then, thegraded object of the filtration is L(4,2)/(2,1)G⊕ L(2,2)/(2,1)F⊗ L(4,2)/(2,2)G, L(3,1)/(2,1)F⊗L(4,2)/(3,1)G, L(3,2)/(2,1)F⊗ L(4,2)/(3,2)G, L(4,1)/(2,1)F⊗ L(4,2)/(4,1)G and L(4,2)/(2,1)F.

(2) The decomposition of L(1,1,1)(F⊕G) corresponds to S3G, F⊗ L(1,1,1)/(1,0,0)G, S2(F)⊗L(1,1,1)/(1,1,0)G and S3F.

In fact, when R contains a field of characteristic zero we have an isomorphism

Lλ/µ(F⊕ G) �⊕

µ⊆γ⊆λLγ/µF⊗ Lλ/γF

(See [2], Proposition (2.3.1).)

2.4 Cauchy Decomposition Formulas for Schur Functors. 25

2.4 Cauchy Decomposition Formulas for Schur Functors.

In this section we explain the relation between Schur and CoSchur functors and thesymmetric and exterior algebra. More precisely, we will construct filtrations of Sk(F⊗ G)and Λk(F ⊗ G) with associated graded objects ⊕|λ|=k LλF ⊗ LλG and ⊕|λ|=k LλF ⊗ KλGrespectively. Moreover, if R is a ring of characteristic zero, these filtrations are direct sumdecompositions of Sk(F⊗G) and Λk(F⊗G). The Cauchy formula for the exterior algebrawill be essential in Chapter 4 to understand the construction of general resolutions ofdeterminantal varieties.

2.4.1 The decomposition of the Symmetric Algebra.

Definition 2.4.1. Let p be a positive integer and let λ = (λ1, . . . , λt) be a partition ofweight k. We define a natural paring 〈, 〉p : ΛpF ⊗ ΛpG → Sp(F ⊗ G) by sending f1 ∧· · · ∧ fp ⊗ g1 ∧ · · · ∧ gp to the p× p determinant (−1)p(p−1)/2 ∑(−1)sg(σ)( fσ(1) ⊗ g1) · · · · ·( fσ(p) ⊗ gp) =: 〈 f1 ∧ · · · ∧ fp, g1 ∧ · · · ∧ gp〉. Formally,

〈 f1 ∧ · · · ∧ fp, g1 ∧ · · · ∧ gp〉p =

∣∣∣∣∣∣∣f1 ⊗ g1 · · · f1 ⊗ gp

......

fp ⊗ g1 · · · fp ⊗ gp

∣∣∣∣∣∣∣ .We extend the above to a pairing 〈, 〉 : ΛλF⊗ΛλG → Sk(F⊗G) by 〈 f1 ∧ · · · ∧ ft, g1 ∧ · · · ∧gt〉 = 〈 f1, g1〉λ1 · · · · · 〈 ft, gt〉, where fi ∈ Λ

λi F and gi ∈ Λλi G for all i ∈ {1, . . . , t}.

For example, 〈, 〉2 : Λ2F ⊗ Λ2G → S2(F ⊗ G) is given by 〈 f1 ∧ f2, g1 ∧ g2〉2 = ( f1 ⊗g1)( f2 ⊗ g2)− ( f1 ⊗ g2)( f2 ⊗ g1), f1, f2 ∈ F, g1, g2 ∈ G.

Definition 2.4.2. Let λ = (λ1, . . . , λt) be a partition of weight k. We define submodules ofSk(F⊗ G),

Mλ(Sk(F⊗ G)) := ∑γ≥λ,|γ|=k

〈ΛγF, ΛγG〉,

Ṁλ(Sk(F⊗ G)) := ∑γ>λ,|γ|=k

〈ΛγF, ΛγG〉.

where 〈ΛγF, ΛγF〉 denotes the image of the pairing 〈, 〉 : ΛγF⊗ΛγG → Sk(F⊗ G).

Note that 〈F⊗ · · · ⊗ F, G⊗ · · · ⊗ G〉 = M(1,...,1)(Sk(F⊗ G)) = Sk(F⊗ G), 1 + · · ·+ 1 =k. We have,

Definition 2.4.3. {Mλ(Sk(F⊗ G) | |λ| = k} define a natural filtration

0 ⊆ M(k)(Sk(F⊗ G)) ⊆ M(k−1,1)(Sk(F⊗ G)) ⊆ · · · ⊆ M(1,...,1)(Sk(F⊗ G))

induced by the lexicographic order ≥ on partitions of weight k.

Directly from the definition of both submodules, {〈XS, YT〉 | S ∈ Tabγ{x1, . . . , xn}, T ∈Tabγ{xn+1, . . . , xn+m}, γ ≥ λ, |γ| = k} form a system of generators of Mλ(Sk(F ⊗ G)),and {〈XS, YT〉 | S ∈ Tabγ{x1, . . . , xn}, T ∈ Tabγ{xn+1, . . . , xn+m}, γ > λ, |γ| = k} spansṀλ(Sk(F⊗ G)).


For convenience we denote Mλ(Sk(F ⊗ G)) by Mλ and Ṁλ(Sk(F ⊗ G)) by Ṁλ. Ourgoal is to show that Mλ/Ṁλ is isomorphic to LλF ⊗ LλG. So first we need to define amorphism βγ from LλF⊗ LλG to Mλ/Ṁλ. In [1] Corollary [I I I.1.2], [I.5] and Proposition[I I I.1.1], using that the paring 〈, 〉 is in fact a restriction of a natural map between extendedR-Hopf algebras, one sees that 〈ωλ(ΛλF), ΛλF〉+ 〈(ΛλF), ωλ(ΛλF)〉 ⊆ Ṁλ. We have,

Proposition 2.4.4. The natural map 〈, 〉 : ΛλF ⊗ ΛλF → Mλ induces a surjective mapβλ : LλF⊗ LλG → Mλ/Ṁλ given by βλ(dλ(X)⊗ dλ(Y)) = 〈X, Y〉, where 〈X, Y〉 denotesthe class of 〈X, Y〉 in the quotient Mλ/Ṁλ.Proof. Since LλF � ΛλF/Im(ωλ), see Definition 2.1.16 and Theorem 2.1.24, it followsfrom 〈ωλ(ΛλF), ΛλF〉+ 〈(ΛλF), ωλ(ΛλF)〉 ⊆ Ṁλ that βλ is well defined. Since Mλ/Ṁλ isgenerated by {〈XS, YT〉 | S ∈ Tabλ{x1, . . . , xn}, T ∈ Tabλ{xn+1, . . . , xn+m}}, it is enoughto see that for each 〈XS, YT〉 of that such there exists X ∈ LλF and Y ∈ LλG such thatβγ(dλ(X)⊗ dλ(Y)) = 〈XS, YT〉. Obviously, XS and YT are the elements we are looking for.�

Remark 2.4.5. As a consequence of the above proposition we obtain {〈XS, YT〉 | S ∈Tabλ{x1, . . . , xn}, T ∈ Tabλ{xn+1, . . . , xn+m} are standard} generates Mλ/Ṁλ.

Corollary 2.4.6. {〈XS, YT〉 | S ∈ Tabγ{x1, . . . , xn}, T ∈ Tabγ{xn+1, . . . , xn+m} are standard,γ ≥ λ, |γ| = k} := Bλ generate Mλ.Proof. Since {Mλ, |λ| = k} is ordered lexicographically, we proceed by induction on λ.We can see easily that the first piece of the decomposition is ΛkF⊗ΛkG = L(k)F⊗ L(k)G.Indeed, M(k) is isomorphic to ΛkF⊗ΛkG, since the pairing 〈, 〉 : ΛkF⊗ΛkG → Sk(F⊗ G)is the natural embedding of ΛkF ⊗ ΛkG in Sk(F ⊗ G). So, the initial case is clear. Con-sider λ > (k), we want to prove that Bλ generate Mλ. Observe that the correspondingpiece of the filtration is Ṁλ ⊆ Mλ. From Proposition 2.4.4, each element of Mλ can bewritten as a sum of an element of Ṁλ and an element generated by {〈XS, YT〉 | S ∈Tabλ{x1, . . . , xn}, T ∈ Tabλ{xn+1, . . . , xn+m} are standard}. The result follows by induc-tion on Ṁλ. �

Theorem 2.4.7. The maps βλ : LλF⊗ LλG → Mλ/Ṁλ are isomorphisms and therefore theassociated graded object of the filtration {Mλ(Sk(F⊗ G) | |λ| = k} is

⊕|λ|=k LλF⊗ LλG.

Proof. Let Bk := {〈XS, YT〉 | S ∈ Tabλ{x1, . . . , xn} is standard, T ∈ Tabλ{xn+1, . . . , xn+m}is standard, |λ| = k}. By Corollary 2.4.6, Bk generates Sk(F⊗ G). Then directly from the

equality ranksrank(Sk(F⊗ G)) = ∑

|λ|=krank(LλF⊗ LλG)

(see [1] Theorem [I I I.1.4]), Bk must be an R-basis of Sk(F⊗ G). This, in turn, implies thatBλ is an R-basis of Mλ and {〈XS, YT〉 | S ∈ Tabγ{x1, . . . , xn}, T ∈ Tabγ{xn+1, . . . , xn+m}are standard, γ > λ, |γ| = k} := Ḃλ is an R-basis of Ṁλ. Consequently, {〈XS, YT〉 |

S ∈ Tabλ{x1, . . . , xn}, T ∈ Tabλ{xn+1, . . . , xn+m} are standard} is an R-basis of Mλ/Ṁλ. Itfollows that βλ send an element basis of LλF⊗ LλG to an element basis of Mλ/Ṁλ. �

Examples 2.4.8. (1) The decomposition of S2(F ⊗ G) corresponds to Λ2F ⊗ Λ2F andS2F⊗ S2G.

2.4 Cauchy Decomposition Formulas for Schur Functors. 27

(2) The three partitions of weight 3 are (3), (2, 1) and (1, 1, 1), then the respective termsof the decomposition of S3(F⊗ G) are Λ3F⊗Λ3F, L(2,1)F⊗ L(2,1)G and S3F⊗ S3G.

The Cauchy formula for Sk(F⊗G) has an important consequence. When R is a ring ofcharacteristic zero the decomposition becomes an isomorphism, more precisely (see [20]Corollary (2.3.3))

Theorem 2.4.9. If R is a ring of characteristic zero, then Sk(F⊗ G) �⊕|λ|=k LλF⊗ LλG.2.4.2 The decomposition of the exterior algebra.

Definition 2.4.10. Let p be a positive integer and let λ = (λ1, . . . , λt) be a partition ofweight k. We define a natural pairing 〈, 〉p : ΛpF⊗DpG → Λp(F⊗G) by induction on p ≥1. For p = 1 we define 〈 f , g〉p = f ⊗ g. For p > 1 we define 〈 f1 ∧ · · · ∧ fp, g

(α1)1 · · · · · g

(αt)t 〉p

byp

∑i=1〈 f1, gi〉 ∧ 〈 f1 ∧ · · · ∧ fp, g

(α1)1 · · · · · g

(αi−1)i · · · · · g

(αt)t 〉

where ∑ti=1 = p and αi ≥ 1 for all i. We define a pairing 〈, 〉 : ΛλF⊗ DλG → Λk(F⊗ G)by the natural extension 〈 f1 ⊗ · · · ⊗ ft, g1 ⊗ · · · ⊗ gt〉 = 〈 f1, g1〉λ1 ∧ · · · ∧ 〈 ft, gt〉λt , wherefi ∈ Λλi F and gi ∈ Dλi G for all i ∈ {1, . . . , t}.

Definition 2.4.11. Let λ = (λ1, . . . , λt) be a partition of weight k. We define submodulesof Λk(F⊗ G)

Mλ(Λk(F⊗ G)) := ∑γ≥λ,|γ|=k

〈ΛγF, DγG〉,

Ṁλ(Λk(F⊗ G)) := ∑γ>λ,|γ|=k

〈ΛγF, DγG〉.

Clearly M(1,...,1) = Λk(F⊗ G), 1 + · · ·+ 1 = k. We have,

Definition 2.4.12. {Mλ(Λk(F⊗ G) | |λ| = k} define a natural filtration

0 ⊆ M(k)(Λk(F⊗ G)) ⊆ M(k−1,1)(Λk(F⊗ G)) ⊆ · · · ⊆ M(1,...,1)(Λk(F⊗ G))

induced by the lexicographic order ≥ on partitions of weight k.

From the above definitions, {〈XS, YT〉 |S∈Tabγ{x1, . . . , xn}, T ∈ Tabγ{xn+1, . . . , xn+m},γ ≥ λ, |γ| = k} and {〈XS, YT〉 | S ∈ Tabγ{x1, . . . , xn}, T ∈ Tabγ{xn+1, . . . , xn+m}, γ >λ, |γ| = k} spans Mλ(Λk(F⊗ G)) and Ṁλ(Λk(F⊗ G)), respectively. Arguing as in Sub-section 2.4.1 we get

Proposition 2.4.13. The natural map 〈, 〉 : ΛλF⊗ΛλG → Mλ induces a surjective R-mapβλ : LλF⊗ KλG → Mλ/Ṁλ.

Lemma 2.4.14. {〈XS, YT〉 | S ∈ Tabγ{x1, . . . , xn} is standard, T ∈ Tabγ{xn+1, . . . , xn+m} isstandard, γ ≥ λ, |γ| = k} spans Mλ.


Theorem 2.4.15. The maps βλ : LλF ⊗ KλG → Mλ/Ṁλ are isomorphisms and thereforethe associated graded object of the filtration {Mλ(Λk(F ⊗ K), |λ| = k} is

⊕|λ|=k LλF ⊗

KλG.Proof. It follows as Theorem 2.4.7 using the equality rank

rank(Λk(F⊗ G)) = ∑|λ|=k

rank(LλF⊗ KλG)

which we can find in [1] Theorem [I I I.2.2]. �

Theorem 2.4.16. If R is a ring of characteristic zero, then Λk(F⊗ G) � ⊕|λ|=k LλF⊗ KλF.Moreover, since KλG � Lλ̃G, Λ

k(F⊗ G) �⊕|λ|=k LλF⊗ Lλ̃G.2.5 The Littlewood-Richardson rule for Schur functors.

The tensor product of Schur functors decomposes into a direct sum of Schur functors,provided that R is a ring of characteristic zero. We finish this exposition about Schur andCoSchur functors presenting a combinatoric algorithm, the Littlewood-Richardson rule,which computes the multiplicities of the factors in the decomposition of LλF ⊗ LµG. Asa corollary of the Littlewood-Richardson rule we will obtain the Pieri formulas for Schurfunctors.

An accurate exposition of the following result, based on representation theory, is foundin [20] Sections [2.2] and [2.3].

Theorem 2.5.1. Let R be a ring of characteristic zero, let F be a free R-module of rank nand let λ and µ be two partitions. Then,

LλF⊗ LµF � ∑|ν|=|λ|+|µ|

u(λ, µ; ν)LνF

where u(λ, µ; ν) denotes the multiplicity of the factor LνF.

The Littlewood-Richardson rule provides an algorithm which computes the multiplic-ities u(λ, µ; ν). The statement and proof of Littlewood-Richardson rule require two com-binatorial notions we present immediately.

2.5.1 The Schensted Process and Words of Yamanouchi.

The following procedure is known as the Schensted process.

Definition 2.5.2. Let λ be a partition, S a totally ordered set, U ∈ Tabλ(S) a standardtableau and p ∈ S. We define p→ U to be the tableau obtained in the following recursivemanner.(1). If p1 = min{U(1, j) | U(1, j) ≥ p} exists, let p → U be the tableau of shape λobtained by replacing of p1 to p. We say this step bumping of p into U.(2). If p1 does not exist we define p → U to be the tableau of shape λ1 = (λ1 + 1, . . . , λq)obtained from U by adjoining a new box to the first row of U with entry p and finish.

2.5 The Littlewood-Richardson rule for Schur functors. 29

If (1) takes place repeat the process with U1 and p = p1, where U1 is the tableauobtained from U by removing the first row, and continue. Finally, p → U is the tableauobtained from U by adjoining a new box to a non empty row or by adjoining a new bottomrow. In the first case p → U is a tableau of shape λ1 = (λ1, . . . , λi + 1, . . . , λn) for some i,and in the second case, λ1 = (λ1, . . . , λn, 1).

Remark 2.5.3. Note that λ1 is a partition. When λ1 = (λ1, . . . , λn, 1) the result is clear.Assume that λ1 = (λ1, . . . , λi + 1, . . . , λn), then in the ith step of the above procedure pdoes not bump any element in the ith row of U. The standardness of U implies U(i −1, j) ≤ U(i, j), for every (i, j) of its diagram. If p = U(i − 1, j) and j ≤ λi, then pi−1 ≤U(i, j) ⇒ ∃min{U(i, j) | U(i, j) ≥ pi−1}, which is a contradiction. Necessarily, j > λi ⇒λi−1 > λi and hence, λi−1 ≥ λi + 1.

Let us to compute p→ U for p = 2 and U = 1 3 42 34

.

(1). p1 = U(1, 2) = 3, p→ U = 1 2 42 34

, λ1 = λ.

Let p = 3 and U1 = 2 34

.

(2). p1 = U(2, 2) = 3, p→ U.

Let p = 3 and U2 = 4 .(3). p1 = U(3, 1) = 4, p→ U = 1 2 4

2 33

, λ1 = λ.

Finish, U → p = 1 2 42 334

and λ1 = (3, 2, 1, 1).

Lemma 2.5.4. The tableau p→ U in Tabλ1(S) is standard.Proof. Row-standardness is clear, we only have to prove column-standardness. It isenough to consider two adjacent rows, i and i + 1, of U in which the step bumping ofp into U took place, otherwise there is nothing to prove since λ1 is a partition. For conve-nience we denote by ui, i = 1, . . . , λi the entries in the ith row and by vi, i = 1, . . . , λi+1 theentries in the (i + 1)th row.(i) Suppose p does bump ut in the ith row and ut does bump vs in the (i + 1)th row. Sinceut−1 ≤ p ≤ ut and vs−1 ≤ ut ≤ vs, we must have s ≤ t and we only have to check thatp ≤ vt. Clearly s ≤ t and the standardness of U implies ut ≤ vs ≤ vt which, in turn,implies p ≤ vt.(ii) Suppose p does bump ut, but ut does not bump any element in the i + 1th row of U.In this case we adjoin the box (i + 1, λi + 1) with the entry ut. Then p ≤ ut

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