research in core partitions 111 reflection...
TRANSCRIPT
![Page 1: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/1.jpg)
Research in Core Partitions 111
Reflection Groups
Today, we will discuss the combinatorics of groups.
◮ Made up of a set of elements W = {w1,w2, . . .}.
◮ Multiplication of two elements w1w2 stays in the group.
![Page 2: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/2.jpg)
Research in Core Partitions 111
Reflection Groups
Today, we will discuss the combinatorics of groups.
◮ Made up of a set of elements W = {w1,w2, . . .}.
◮ Multiplication of two elements w1w2 stays in the group.
◮ ALTHOUGH, it is not the case that w1w2 = w2w1.
![Page 3: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/3.jpg)
Research in Core Partitions 111
Reflection Groups
Today, we will discuss the combinatorics of groups.
◮ Made up of a set of elements W = {w1,w2, . . .}.
◮ Multiplication of two elements w1w2 stays in the group.
◮ ALTHOUGH, it is not the case that w1w2 = w2w1.
◮ There is an identity element (id) & Every element has an inverse.
![Page 4: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/4.jpg)
Research in Core Partitions 111
Reflection Groups
Today, we will discuss the combinatorics of groups.
◮ Made up of a set of elements W = {w1,w2, . . .}.
◮ Multiplication of two elements w1w2 stays in the group.
◮ ALTHOUGH, it is not the case that w1w2 = w2w1.
◮ There is an identity element (id) & Every element has an inverse.
◮ Think: (Non-zero real numbers) or (invertible n× n matrices.)
![Page 5: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/5.jpg)
Research in Core Partitions 111
Reflection Groups
Today, we will discuss the combinatorics of groups.
◮ Made up of a set of elements W = {w1,w2, . . .}.
◮ Multiplication of two elements w1w2 stays in the group.
◮ ALTHOUGH, it is not the case that w1w2 = w2w1.
◮ There is an identity element (id) & Every element has an inverse.
◮ Think: (Non-zero real numbers) or (invertible n× n matrices.)
We will talk about reflection groups. (With nice pictures)
◮ W is generated by a set of generators S = {s1, s2, . . . , sk}.
◮ Along with a set of relations.
![Page 6: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/6.jpg)
Research in Core Partitions 111
Reflection Groups
Today, we will discuss the combinatorics of groups.
◮ Made up of a set of elements W = {w1,w2, . . .}.
◮ Multiplication of two elements w1w2 stays in the group.
◮ ALTHOUGH, it is not the case that w1w2 = w2w1.
◮ There is an identity element (id) & Every element has an inverse.
◮ Think: (Non-zero real numbers) or (invertible n× n matrices.)
We will talk about reflection groups. (With nice pictures)
◮ W is generated by a set of generators S = {s1, s2, . . . , sk}.
◮ Every w ∈W can be written as a product of generators.
◮ Along with a set of relations.
![Page 7: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/7.jpg)
Research in Core Partitions 111
Reflection Groups
Today, we will discuss the combinatorics of groups.
◮ Made up of a set of elements W = {w1,w2, . . .}.
◮ Multiplication of two elements w1w2 stays in the group.
◮ ALTHOUGH, it is not the case that w1w2 = w2w1.
◮ There is an identity element (id) & Every element has an inverse.
◮ Think: (Non-zero real numbers) or (invertible n× n matrices.)
We will talk about reflection groups. (With nice pictures)
◮ W is generated by a set of generators S = {s1, s2, . . . , sk}.
◮ Every w ∈W can be written as a product of generators.
◮ Along with a set of relations.
◮ These are rules to convert between expressions.◮ s2
i = id. —and— (si sj)power = id.
![Page 8: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/8.jpg)
Research in Core Partitions 111
Reflection Groups
Today, we will discuss the combinatorics of groups.
◮ Made up of a set of elements W = {w1,w2, . . .}.
◮ Multiplication of two elements w1w2 stays in the group.
◮ ALTHOUGH, it is not the case that w1w2 = w2w1.
◮ There is an identity element (id) & Every element has an inverse.
◮ Think: (Non-zero real numbers) or (invertible n× n matrices.)
We will talk about reflection groups. (With nice pictures)
◮ W is generated by a set of generators S = {s1, s2, . . . , sk}.
◮ Every w ∈W can be written as a product of generators.
◮ Along with a set of relations.
◮ These are rules to convert between expressions.◮ s2
i = id. —and— (si sj)power = id.
For example, w = s3s2s1s1s2s4 = s3s2ids2s4 = s3ids4 = s3s4
![Page 9: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/9.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
![Page 10: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/10.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
![Page 11: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/11.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
s
![Page 12: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/12.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
t
s
![Page 13: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/13.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
t
ts
s
st
![Page 14: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/14.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
t
ts
s
st
◮ When the angle between Hs and Ht is π
3, relation is (st)3 = id.
![Page 15: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/15.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
t
ts
s
st
◮ When the angle between Hs and Ht is π
3, relation is (st)3 = id.
◮ The group depends on the placement of the hyperplanes. |S | = 6.
![Page 16: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/16.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
◮ When the angle between Hs and Ht is π
4, relation is (st)4 = id.
◮ The group depends on the placement of the hyperplanes. |S | = 8.
![Page 17: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/17.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
◮ When the angle between Hs and Ht is π
5, relation is (st)5 = id.
◮ The group depends on the placement of the hyperplanes. |S |=10.
![Page 18: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/18.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
◮ When the angle between Hs and Ht is π
6, relation is (st)6 = id.
◮ The group depends on the placement of the hyperplanes. |S |=12.
![Page 19: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/19.jpg)
Research in Core Partitions 112
Reflection Groups
◮ The action of multiplying (on the left) by a generator s
corresponds to a reflection across a hyperplane Hs . (s2i = id)
s
t
◮ When the angle between Hs and Ht is π
n, relation is (st)n = id.
◮ The group depends on the placement of the hyperplanes. |S |=2n.
![Page 20: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/20.jpg)
Research in Core Partitions 113
Infinite Reflection Groups
An infinite reflection group: the affine permutations S̃n.
s1
s2
![Page 21: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/21.jpg)
Research in Core Partitions 113
Infinite Reflection Groups
An infinite reflection group: the affine permutations S̃n.
s1
s2
![Page 22: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/22.jpg)
Research in Core Partitions 113
Infinite Reflection Groups
An infinite reflection group: the affine permutations S̃n.
s1
s2
![Page 23: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/23.jpg)
Research in Core Partitions 113
Infinite Reflection Groups
An infinite reflection group: the affine permutations S̃n.
◮ Add a new generator s0 and a new affine hyperplane H0.
s1
s2
s0
![Page 24: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/24.jpg)
Research in Core Partitions 113
Infinite Reflection Groups
An infinite reflection group: the affine permutations S̃n.
◮ Add a new generator s0 and a new affine hyperplane H0.
s1
s2
s0
![Page 25: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/25.jpg)
Research in Core Partitions 113
Infinite Reflection Groups
An infinite reflection group: the affine permutations S̃n.
◮ Add a new generator s0 and a new affine hyperplane H0.
s1
s2
s0
![Page 26: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/26.jpg)
Research in Core Partitions 113
Infinite Reflection Groups
An infinite reflection group: the affine permutations S̃n.
◮ Add a new generator s0 and a new affine hyperplane H0.
s1
s2
s0
![Page 27: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/27.jpg)
Research in Core Partitions 113
Infinite Reflection Groups
An infinite reflection group: the affine permutations S̃n.
◮ Add a new generator s0 and a new affine hyperplane H0.
s1
s2
s0
Elements generated by {s0, s1, s2} correspond to alcoves here.
![Page 28: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/28.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
H1
H2
H0
![Page 29: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/29.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
◮ Geometry. Point to the alcove.
H1
H2
H0
![Page 30: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/30.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
◮ Geometry. Point to the alcove.
◮ Alcove coordinates. Keep track ofhow many hyperplanes of each typeyou have crossed to get to your alcove. H1
H2
H0
Coordinates:
3 1
1
![Page 31: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/31.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
◮ Geometry. Point to the alcove.
◮ Alcove coordinates. Keep track ofhow many hyperplanes of each typeyou have crossed to get to your alcove.
◮ Word. Write the element as a(short) product of generators.
H1
H2
H0
Coordinates:
3 1
1
Word: s0s1s2s1s0
![Page 32: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/32.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
◮ Geometry. Point to the alcove.
◮ Alcove coordinates. Keep track ofhow many hyperplanes of each typeyou have crossed to get to your alcove.
◮ Word. Write the element as a(short) product of generators.
◮ Permutation. Similar to writingfinite permutations as 312.
H1
H2
H0
Coordinates:
3 1
1
Word: s0s1s2s1s0
Permutation:
(−3, 2, 7)
![Page 33: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/33.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
◮ Geometry. Point to the alcove.
◮ Alcove coordinates. Keep track ofhow many hyperplanes of each typeyou have crossed to get to your alcove.
◮ Word. Write the element as a(short) product of generators.
◮ Permutation. Similar to writingfinite permutations as 312.
◮ Abacus diagram. Columns of numbers.
H1
H2
H0
Abacus diagram:
10
7
4
1
-2
-5
-8
11
8
5
2
-1
-4
-7
12
9
6
3
0
-3
-6
![Page 34: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/34.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
◮ Geometry. Point to the alcove.
◮ Alcove coordinates. Keep track ofhow many hyperplanes of each typeyou have crossed to get to your alcove.
◮ Word. Write the element as a(short) product of generators.
◮ Permutation. Similar to writingfinite permutations as 312.
◮ Abacus diagram. Columns of numbers.
◮ Core partition. Hook length condition.
H1
H2
H0
Core partition:0 1 2 0
2 0
1
0
![Page 35: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/35.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
◮ Geometry. Point to the alcove.
◮ Alcove coordinates. Keep track ofhow many hyperplanes of each typeyou have crossed to get to your alcove.
◮ Word. Write the element as a(short) product of generators.
◮ Permutation. Similar to writingfinite permutations as 312.
◮ Abacus diagram. Columns of numbers.
◮ Core partition. Hook length condition.
◮ Bounded partition. Part size bounded.
H1
H2
H0
Core partition:0 1 2 0
2 0
1
0
Bounded partition:0 1
2
1
0
![Page 36: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/36.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
◮ Geometry. Point to the alcove.
◮ Alcove coordinates. Keep track ofhow many hyperplanes of each typeyou have crossed to get to your alcove.
◮ Word. Write the element as a(short) product of generators.
◮ Permutation. Similar to writingfinite permutations as 312.
◮ Abacus diagram. Columns of numbers.
◮ Core partition. Hook length condition.
◮ Bounded partition. Part size bounded.
◮ Others! Lattice path, order ideal, etc.
H1
H2
H0
Core partition:0 1 2 0
2 0
1
0
Bounded partition:0 1
2
1
0
![Page 37: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/37.jpg)
Research in Core Partitions 114
Combinatorics of affine permutations
Many ways to reference elements in S̃n.
◮ Geometry. Point to the alcove.
◮ Alcove coordinates. Keep track ofhow many hyperplanes of each typeyou have crossed to get to your alcove.
◮ Word. Write the element as a(short) product of generators.
◮ Permutation. Similar to writingfinite permutations as 312.
◮ Abacus diagram. Columns of numbers.
◮ Core partition. Hook length condition.
◮ Bounded partition. Part size bounded.
◮ Others! Lattice path, order ideal, etc.
They all play nicely with each other.
H1
H2
H0
Core partition:0 1 2 0
2 0
1
0
Bounded partition:0 1
2
1
0
![Page 38: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/38.jpg)
Research in Core Partitions 115
An abacus model for affine permutations
(James and Kerber, 1981) Given an affine permutation [w1, . . . ,wn],
◮ Place integers in n runners.
17
13
9
5
1
-3
-7
-11
-15
18
14
10
6
2
-2
-6
-10
-14
19
15
11
7
3
-1
-5
-9
-13
20
16
12
8
4
0
-4
-8
-12
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Research in Core Partitions 115
An abacus model for affine permutations
(James and Kerber, 1981) Given an affine permutation [w1, . . . ,wn],
◮ Place integers in n runners.
◮ Circled: beads. Empty: gaps
17
13
9
5
1
-3
-7
-11
-15
18
14
10
6
2
-2
-6
-10
-14
19
15
11
7
3
-1
-5
-9
-13
20
16
12
8
4
0
-4
-8
-12
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Research in Core Partitions 115
An abacus model for affine permutations
(James and Kerber, 1981) Given an affine permutation [w1, . . . ,wn],
◮ Place integers in n runners.
◮ Circled: beads. Empty: gaps
◮ Create an abacus where eachrunner has a lowest bead at wi .
Example: [−4,−3, 7, 10]17
13
9
5
1
- 3
-7
-11
-15
18
14
10
6
2
-2
-6
-10
-14
19
15
11
7
3
-1
-5
-9
-13
20
16
12
8
4
0
- 4
-8
-12
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Research in Core Partitions 115
An abacus model for affine permutations
(James and Kerber, 1981) Given an affine permutation [w1, . . . ,wn],
◮ Place integers in n runners.
◮ Circled: beads. Empty: gaps
◮ Create an abacus where eachrunner has a lowest bead at wi .
Example: [−4,−3, 7, 10]17
13
9
5
1
- 3
-7
-11
-15
18
14
10
6
2
-2
-6
-10
-14
19
15
11
7
3
-1
-5
-9
-13
20
16
12
8
4
0
- 4
-8
-12
◮ Generators act nicely.
◮ si interchanges runners i ↔ i + 1.
◮ s0 interchanges runners 1 and n (with shifts)
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Research in Core Partitions 115
An abacus model for affine permutations
(James and Kerber, 1981) Given an affine permutation [w1, . . . ,wn],
◮ Place integers in n runners.
◮ Circled: beads. Empty: gaps
◮ Create an abacus where eachrunner has a lowest bead at wi .
Example: [−4,−3, 7, 10]17
13
9
5
1
- 3
-7
-11
-15
18
14
10
6
2
-2
-6
-10
-14
19
15
11
7
3
-1
-5
-9
-13
20
16
12
8
4
0
- 4
-8
-12
s1→
17
13
9
5
1
-3
-7
-11
-15
18
14
10
6
2
- 2
-6
-10
-14
19
15
11
7
3
-1
-5
-9
-13
20
16
12
8
4
0
- 4
-8
-12
◮ Generators act nicely.
◮ si interchanges runners i ↔ i + 1. (s1 : 1↔ 2)
◮ s0 interchanges runners 1 and n (with shifts)
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Research in Core Partitions 115
An abacus model for affine permutations
(James and Kerber, 1981) Given an affine permutation [w1, . . . ,wn],
◮ Place integers in n runners.
◮ Circled: beads. Empty: gaps
◮ Create an abacus where eachrunner has a lowest bead at wi .
Example: [−4,−3, 7, 10]17
13
9
5
1
- 3
-7
-11
-15
18
14
10
6
2
-2
-6
-10
-14
19
15
11
7
3
-1
-5
-9
-13
20
16
12
8
4
0
- 4
-8
-12
s1→
17
13
9
5
1
-3
-7
-11
-15
18
14
10
6
2
- 2
-6
-10
-14
19
15
11
7
3
-1
-5
-9
-13
20
16
12
8
4
0
- 4
-8
-12
s0→
17
13
9
5
1
- 3
-7
-11
-15
18
14
10
6
2
- 2
-6
-10
-14
19
15
11
7
3
-1
-5
-9
-13
20
16
12
8
4
0
-4
-8
-12
◮ Generators act nicely.
◮ si interchanges runners i ↔ i + 1. (s1 : 1↔ 2)
◮ s0 interchanges runners 1 and n (with shifts) (s0 : 1shift↔ 4)
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Research in Core Partitions 116
Core partitions
For an integer partition λ = (λ1, . . . , λk) drawn as a Young diagram,
The hook length of a box is # boxes below and to the right.
10 9 6 5 2 1
7 6 3 2
6 5 2 1
3 2
2 1
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Research in Core Partitions 116
Core partitions
For an integer partition λ = (λ1, . . . , λk) drawn as a Young diagram,
The hook length of a box is # boxes below and to the right.
10 9 6 5 2 1
7 6 3 2
6 5 2 1
3 2
2 1
An n-core is a partition with no boxes of hook length dividing n.
Example. λ is a 4-core, 8-core, 11-core, 12-core, etc.λ is NOT a 1-, 2-, 3-, 5-, 6-, 7-, 9-, or 10-core.
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Research in Core Partitions 117
Core partition interpretation for affine permutations
Bijection: {abaci} ←→ {n-cores}
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Research in Core Partitions 117
Core partition interpretation for affine permutations
Bijection: {abaci} ←→ {n-cores}
Rule: Read the boundary steps of λ from the abacus:
◮ A bead ↔ vertical step ◮ A gap ↔ horizontal step
13
9
5
1
-3
-7
-11
14
10
6
2
-2
-6
-10
15
11
7
3
-1
-5
-9
16
12
8
4
0
-4
-8
←→
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Research in Core Partitions 117
Core partition interpretation for affine permutations
Bijection: {abaci} ←→ {n-cores}
Rule: Read the boundary steps of λ from the abacus:
◮ A bead ↔ vertical step ◮ A gap ↔ horizontal step
13
9
5
1
-3
-7
-11
14
10
6
2
-2
-6
-10
15
11
7
3
-1
-5
-9
16
12
8
4
0
-4
-8
←→
Fact: This is a bijection!
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Research in Core Partitions 118
Action of generators on the core partition
◮ Label the boxes of λ with residues.
◮ si acts by adding or removing boxeswith residue i .
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
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Research in Core Partitions 118
Action of generators on the core partition
◮ Label the boxes of λ with residues.
◮ si acts by adding or removing boxeswith residue i .
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
Example. λ = (5, 3, 3, 1, 1)
◮ has removable 0 boxes
◮ has addable 1, 2, 3 boxes.
s2
0 1 2 3 0 13 0 1 2 3 02 3 0 1 2 31 2 3 0 1 20 1 2 3 0 13 0 1 2 3 0
s0→
0 1 2 3 0 13 0 1 2 3 02 3 0 1 2 31 2 3 0 1 20 1 2 3 0 13 0 1 2 3 0
s1 ↓ ց0 1 2 3 0 13 0 1 2 3 02 3 0 1 2 31 2 3 0 1 20 1 2 3 0 13 0 1 2 3 0
0 1 2 3 0 13 0 1 2 3 02 3 0 1 2 31 2 3 0 1 20 1 2 3 0 13 0 1 2 3 0
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Research in Core Partitions 118
Action of generators on the core partition
◮ Label the boxes of λ with residues.
◮ si acts by adding or removing boxeswith residue i .
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
Example. λ = (5, 3, 3, 1, 1)
◮ has removable 0 boxes
◮ has addable 1, 2, 3 boxes.
s2
0 1 2 3 0 13 0 1 2 3 02 3 0 1 2 31 2 3 0 1 20 1 2 3 0 13 0 1 2 3 0
s0→
0 1 2 3 0 13 0 1 2 3 02 3 0 1 2 31 2 3 0 1 20 1 2 3 0 13 0 1 2 3 0
s1 ↓ ց0 1 2 3 0 13 0 1 2 3 02 3 0 1 2 31 2 3 0 1 20 1 2 3 0 13 0 1 2 3 0
0 1 2 3 0 13 0 1 2 3 02 3 0 1 2 31 2 3 0 1 20 1 2 3 0 13 0 1 2 3 0
Idea: We can use this tofigure out a word for w .
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Research in Core Partitions 119
Finding a word for an affine permutation.
Example: The word in S4
corresponding to λ = (6, 4, 4, 2, 2):
s1s0s2s1s3s2s0s3s1s0
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s1→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s0→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s2→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s1→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s3→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s2→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s0→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s3→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s1→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
s0→
0 1 2 3 0 1
3 0 1 2 3 0
2 3 0 1 2 3
1 2 3 0 1 2
0 1 2 3 0 1
3 0 1 2 3 0
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Research in Core Partitions 120
The bijection between cores and alcoves
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores?
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores?
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.How many simultaneous 4/5-cores?
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.How many simultaneous 4/5-cores? 14.
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.How many simultaneous 4/5-cores? 14.How many simultaneous 5/6-cores? 42.
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.How many simultaneous 4/5-cores? 14.How many simultaneous 5/6-cores? 42.How many simultaneous n/(n + 1)-cores?
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.How many simultaneous 4/5-cores? 14.How many simultaneous 5/6-cores? 42.How many simultaneous n/(n + 1)-cores? Cn!
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.How many simultaneous 4/5-cores? 14.How many simultaneous 5/6-cores? 42.How many simultaneous n/(n + 1)-cores? Cn!
Jaclyn Anderson proved that the number of s/t-cores is 1s+t
(s+ts
).
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.How many simultaneous 4/5-cores? 14.How many simultaneous 5/6-cores? 42.How many simultaneous n/(n + 1)-cores? Cn!
Jaclyn Anderson proved that the number of s/t-cores is 1s+t
(s+ts
).
The number of 3/7-cores is 110
(103
)= 1
1010·9·83·2·1
= 12.
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.How many simultaneous 4/5-cores? 14.How many simultaneous 5/6-cores? 42.How many simultaneous n/(n + 1)-cores? Cn!
Jaclyn Anderson proved that the number of s/t-cores is 1s+t
(s+ts
).
The number of 3/7-cores is 110
(103
)= 1
1010·9·83·2·1
= 12.
Fishel–Vazirani proved an alcove interpretation of n/(mn+1)-cores.
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Research in Core Partitions 121
Simultaneous core partitions
How many partitions are both 2-cores and 3-cores? 2.
0 10 1 2 0
2 0
0 1 2 0 1 2
2 0 1 2
1 2
0 1 2 0 1 2 0 1
2 0 1 2 0 1
1 2 0 1
0 1
0
2
0 1 2
2
1
0 1 2 0 1
2 0 1
1
0
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
2
0 1
2 0
1
0
0 1 2 0
2 0
1 2
0
2
0 1 2 0 1 2
2 0 1 2
1 2
0 1
2
1
0 1 2
2 0 1
1 2
0 1
2
1
0 1 2 0 1
2 0 1
1 2 0
0 1
2 0
1
0
0 1 2 0
2 0 1 2
1 2 0
0 1 2
2 0
1 2
0
2
00 1 2
2
0 1 2 0 1
2 0 1
1
0 1 2 0 1 2 0
2 0 1 2 0
1 2 0
0
0 1
2
1
0 1 2 0
2 0
1
0
0 1 2 0 1 2
2 0 1 2
1 2
0
2
0 1 2
2 0
1 2
0
2
0 1 2 0 1
2 0 1
1 2
0 1
2
1
0 1 2 0
2 0 1
1 2 0
0 1
2 0
1
0
How many partitions are both 3-cores and 4-cores? 5.How many simultaneous 4/5-cores? 14.How many simultaneous 5/6-cores? 42.How many simultaneous n/(n + 1)-cores? Cn!
Jaclyn Anderson proved that the number of s/t-cores is 1s+t
(s+ts
).
The number of 3/7-cores is 110
(103
)= 1
1010·9·83·2·1
= 12.
Fishel–Vazirani proved an alcove interpretation of n/(mn+1)-cores.
![Page 67: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/67.jpg)
Research in Core Partitions 122
To Mathematica!
◮ Higher dimensions!
◮ More pretty pictures!
◮ Animations in the Research section of my webpage.
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Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
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Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
◮ Yes! Involves self-conjugate partitions.
22
1581
-6
-13
-20
23
1692
-5
-12
-19
24
17
103
-4
-11
-18
25
18
114
-3
-10
-17
26
19
125
- 2
-9
-16
27
20
136
- 1
-8
-15
←→
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Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
◮ Yes! Involves self-conjugate partitions.
◮ Article written (28 pp), submitted.
◮ Joint with Brant Jones, James Madison University.
22
1581
-6
-13
-20
23
1692
-5
-12
-19
24
17
103
-4
-11
-18
25
18
114
-3
-10
-17
26
19
125
- 2
-9
-16
27
20
136
- 1
-8
-15
←→
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Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
◮ Yes! Involves self-conjugate partitions.
◮ Article written (28 pp), submitted.
◮ Joint with Brant Jones, James Madison University.
22
1581
-6
-13
-20
23
1692
-5
-12
-19
24
17
103
-4
-11
-18
25
18
114
-3
-10
-17
26
19
125
- 2
-9
-16
27
20
136
- 1
-8
-15
←→
⋆ What numerical properties do self-conjugate core partitions have?
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Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
◮ Yes! Involves self-conjugate partitions.
◮ Article written (28 pp), submitted.
◮ Joint with Brant Jones, James Madison University.
22
1581
-6
-13
-20
23
1692
-5
-12
-19
24
17
103
-4
-11
-18
25
18
114
-3
-10
-17
26
19
125
- 2
-9
-16
27
20
136
- 1
-8
-15
←→
⋆ What numerical properties do self-conjugate core partitions have?
◮ Working with Rishi Nath, York College, CUNY.
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Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
◮ Yes! Involves self-conjugate partitions.
◮ Article written (28 pp), submitted.
◮ Joint with Brant Jones, James Madison University.
22
1581
-6
-13
-20
23
1692
-5
-12
-19
24
17
103
-4
-11
-18
25
18
114
-3
-10
-17
26
19
125
- 2
-9
-16
27
20
136
- 1
-8
-15
←→
⋆ What numerical properties do self-conjugate core partitions have?
◮ Working with Rishi Nath, York College, CUNY.
◮ In progress. We found some impressive numerical conjectures.
![Page 74: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/74.jpg)
Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
◮ Yes! Involves self-conjugate partitions.
◮ Article written (28 pp), submitted.
◮ Joint with Brant Jones, James Madison University.
22
1581
-6
-13
-20
23
1692
-5
-12
-19
24
17
103
-4
-11
-18
25
18
114
-3
-10
-17
26
19
125
- 2
-9
-16
27
20
136
- 1
-8
-15
←→
⋆ What numerical properties do self-conjugate core partitions have?
◮ Working with Rishi Nath, York College, CUNY.
◮ In progress. We found some impressive numerical conjectures.
◮ There are more (s.c. t+2-cores of n) than (s.c. t-cores of n).
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Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
◮ Yes! Involves self-conjugate partitions.
◮ Article written (28 pp), submitted.
◮ Joint with Brant Jones, James Madison University.
⋆ What numerical properties do self-conjugate core partitions have?
◮ Working with Rishi Nath, York College, CUNY.
◮ In progress. We found some impressive numerical conjectures.
◮ There are more (s.c. t+2-cores of n) than (s.c. t-cores of n).4-cores of 22 6-cores of 22 8-cores of 22
13 11 6 5 4 3 1
11 9 4 3 2 1
6 4
5 3
4 2
3 1
1
17 11 8 7 5 4 3 2 1
11 5 2 1
8 2
7 1
5
4
3
2
1
13 9 8 7 3 2 1
9 5 4 3
8 4 3 2
7 3 2 1
3
2
1
19 11 9 7 6 5 4 3 2 1
11 3 1
9 1
7
6
5
4
3
2
1
13 11 6 5 4 3 1
11 9 4 3 2 1
6 4
5 3
4 2
3 1
1
15 11 7 6 5 3 2 1
11 7 3 2 1
7 3
6 2
5 1
3
2
111 9 7 6 3 1
9 7 5 4 1
7 5 3 2
6 4 2 1
3 1
1
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Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
◮ Yes! Involves self-conjugate partitions.
◮ Article written (28 pp), submitted.
◮ Joint with Brant Jones, James Madison University.
⋆ What numerical properties do self-conjugate core partitions have?
◮ Working with Rishi Nath, York College, CUNY.
◮ In progress. We found some impressive numerical conjectures.
◮ There are more (s.c. t+2-cores of n) than (s.c. t-cores of n).
⋆ What is the average size of an s/t-core partition?
◮ In progress. We “know” the answer, but we have to prove it!
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Research in Core Partitions 123
Research Questions
⋆ Can we extend combinatorial interps to other reflection groups?
◮ Yes! Involves self-conjugate partitions.
◮ Article written (28 pp), submitted.
◮ Joint with Brant Jones, James Madison University.
⋆ What numerical properties do self-conjugate core partitions have?
◮ Working with Rishi Nath, York College, CUNY.
◮ In progress. We found some impressive numerical conjectures.
◮ There are more (s.c. t+2-cores of n) than (s.c. t-cores of n).
⋆ What is the average size of an s/t-core partition?
◮ In progress. We “know” the answer, but we have to prove it!
◮ Working with Drew Armstrong, University of Miami.
![Page 78: Research in Core Partitions 111 Reflection Groupspeople.qc.cuny.edu/faculty/christopher.hanusa/... · Research in Core Partitions 111 Reflection Groups Today, we will discuss the](https://reader034.vdocuments.mx/reader034/viewer/2022050414/5f8aa3fbff0ef7656f3205ec/html5/thumbnails/78.jpg)
Research in Core Partitions 124
Course Evaluation
Please comment on:
◮ Prof. Chris’s effectiveness as a teacher.
◮ Prof. Chris’s contribution to your learning.
◮ The course material: What you enjoyed and/or found challenging.
◮ The book: Pros and Cons.
◮ Is there anything you would change about the course?
◮ Is there anything else Prof. Chris should know?
Place completed evaluations in the provided folder.
I will be in my office, Kiely Tower, Room 409.