[PPT] - Applications of abacus diagrams: Simultaneous core partitions, PowerPoint Presentation

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Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic

Christopher R. H. Hanusa Queens College, CUNY

Joint work with Brant Jones, James Madison University and Drew Armstrong, University of Miami people.qc.cuny.edu/chanusa > Talks

SLIDE 2

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Partitions

The Young diagram of λ = (λ1, . . . , λk) has λi boxes in row i. Partition Self-conjugate partition

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 1 / 15

SLIDE 3

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Partitions

The Young diagram of λ = (λ1, . . . , λk) has λi boxes in row i. (James, Kerber) Create an abacus diagram from the boundary of λ. Abacus: Function a : Z → {•, }. Partition Self-conjugate partition

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 1 / 15

SLIDE 4

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Partitions

The Young diagram of λ = (λ1, . . . , λk) has λi boxes in row i. (James, Kerber) Create an abacus diagram from the boundary of λ. Abacus: Function a : Z → {•, }. Partitions correspond to abacus diagrams.

9 -8 -7 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7 8 9

Partition Self-conjugate partition

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 1 / 15

SLIDE 5

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Partitions

The Young diagram of λ = (λ1, . . . , λk) has λi boxes in row i. (James, Kerber) Create an abacus diagram from the boundary of λ. Abacus: Function a : Z → {•, }. (Equivalence class...) Partitions correspond to abacus diagrams.

6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7 8 9 10 11 12

Partition Self-conjugate partition

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 1 / 15

SLIDE 6

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Partitions

The Young diagram of λ = (λ1, . . . , λk) has λi boxes in row i. (James, Kerber) Create an abacus diagram from the boundary of λ. Abacus: Function a : Z → {•, }. (Equivalence class...) Partitions correspond to abacus diagrams.

9 -8 -7 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7 8 9

Partition Self-conjugate partition Self-conjugate partitions correspond to anti-symmetric abaci.

8
7
6
5
4
3
2
1

1 2 3 4 5 6 7 8 9

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 1 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions

The hook length of a box = # boxes below + # boxes to right + box λ is a t-core if no boxes have hook length t. t-core partition

10 6 5 2 1 7 3 2 6 2 1 3 2 1

t-flush abacus

_ _ _ _ _ _ _ _

Self-conj. t-core partition

13 9 7 5 3 2 1 9 5 3 1 7 3 1 5 1 3 2 1

(Discuss defining beads, reading off hooks....)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 2 / 15

SLIDE 8

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions

The hook length of a box = # boxes below + # boxes to right + box λ is a t-core if no boxes have hook length t ← → t-flush abacus t-core partition

10 6 5 2 1 7 3 2 6 2 1 3 2 1

t-flush abacus

_ _ _ _ _ _ _ _

Self-conj. t-core partition

13 9 7 5 3 2 1 9 5 3 1 7 3 1 5 1 3 2 1

(Discuss defining beads, reading off hooks....)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 2 / 15

SLIDE 9

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions

The hook length of a box = # boxes below + # boxes to right + box λ is a t-core if no boxes have hook length t ← → t-flush abacus t-core partition

10 6 5 2 1 7 3 2 6 2 1 3 2 1

t-flush abacus

5 -4 -3 -2 -1

1 2 3 4 5 6 7 8 9 10 11 12 13

Self-conj. t-core partition

13 9 7 5 3 2 1 9 5 3 1 7 3 1 5 1 3 2 1

(Discuss defining beads, reading off hooks....)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 2 / 15

SLIDE 10

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions

The hook length of a box = # boxes below + # boxes to right + box λ is a t-core if no boxes have hook length t ← → t-flush abacus t-core partition

10 6 5 2 1 7 3 2 6 2 1 3 2 1

t-flush abacus (in runners)

5 -4 -3 -2 -1

1 2 3 4 5 6 7 8 9 10 11 12 13

8 4

4
8

9 5 1

3
7

10 6 2

2
6

11 7 3

1
5

9 5 1

3
7

10 6 2

2
6

11 7 3

1
5

12 8 4

4

Normalized Balanced Self-conj. t-core partition

13 9 7 5 3 2 1 9 5 3 1 7 3 1 5 1 3 2 1

(Discuss defining beads, reading off hooks....)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 2 / 15

SLIDE 11

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions

The hook length of a box = # boxes below + # boxes to right + box λ is a t-core if no boxes have hook length t ← → t-flush abacus t-core partition

10 6 5 2 1 7 3 2 6 2 1 3 2 1

t-flush abacus (in runners)

5 -4 -3 -2 -1

1 2 3 4 5 6 7 8 9 10 11 12 13

8 4

4
8

9 5 1

3
7

10 6 2

2
6

11 7 3

1
5

9 5 1

3
7

10 6 2

2
6

11 7 3

1
5

12 8 4

4

Normalized Balanced Self-conj. t-core partition

13 9 7 5 3 2 1 9 5 3 1 7 3 1 5 1 3 2 1

t-flush antisymmetric abacus

9 5 1

3
7

10 6 2

2
6

11 7 3

1
5

12 8 4

4

Antisymmetry about t/t + 1. (Discuss defining beads, reading off hooks....)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 2 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Simultaneity

Of interest: Partitions that are both s-core and t-core. (s, t) = 1

◮ Abaci that are both s-flush and t-flush.

(s, t)-core partitions

9 6 5 3 2 1 5 2 1 2 1

Self-conj. (s, t)-core partitions

9 6 4 2 1 6 3 1 4 1 2 1 Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 3 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Simultaneity

Of interest: Partitions that are both s-core and t-core. (s, t) = 1

◮ Abaci that are both s-flush and t-flush.

There are infinitely many (self-conjugate) t-core partitions. (s, t)-core partitions

9 6 5 3 2 1 5 2 1 2 1

Self-conj. (s, t)-core partitions

9 6 4 2 1 6 3 1 4 1 2 1 Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 3 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Simultaneity

Of interest: Partitions that are both s-core and t-core. (s, t) = 1

◮ Abaci that are both s-flush and t-flush.

There are infinitely many (self-conjugate) t-core partitions. (s, t)-core partitions

9 6 5 3 2 1 5 2 1 2 1

(Anderson, 2002): # (s, t)-core partitions

1 s+t

s+t

s

Self-conj. (s, t)-core partitions

9 6 4 2 1 6 3 1 4 1 2 1 Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 3 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Simultaneity

Of interest: Partitions that are both s-core and t-core. (s, t) = 1

◮ Abaci that are both s-flush and t-flush.

There are infinitely many (self-conjugate) t-core partitions. (s, t)-core partitions

9 6 5 3 2 1 5 2 1 2 1

(Anderson, 2002): # (s, t)-core partitions

1 s+t

s+t

s

Self-conj. (s, t)-core partitions

9 6 4 2 1 6 3 1 4 1 2 1

(Ford, Mai, Sze, 2009): # self-conj. (s, t)-core partitions s′+t′

s′

where s′ =

s

2

and t′ =

t

2

Applications of abacus diagrams:

Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 3 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47 s-c t-cores arise in rep. thy. of An.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47 s-c t-cores arise in rep. thy. of An.

Readable survey by Kleshchev ’10.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47 s-c t-cores arise in rep. thy. of An.

Readable survey by Kleshchev ’10.

Numerical properties: ct(n) = # of t-core partitions of n.

n≥0

ct(n)qn =

n≥1

(1 − qnt)t 1 − qn (↑ Olsson ’76)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

SLIDE 20

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47 s-c t-cores arise in rep. thy. of An.

Readable survey by Kleshchev ’10.

Numerical properties: ct(n) = # of t-core partitions of n.

n≥0

ct(n)qn =

n≥1

(1 − qnt)t 1 − qn (↑ Olsson ’76) (↓ Granville-Ono ’96)

Positivity. ct(n) > 0 (t ≥ 4).

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

SLIDE 21

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47 s-c t-cores arise in rep. thy. of An.

Readable survey by Kleshchev ’10.

Numerical properties: ct(n) = # of t-core partitions of n.

n≥0

ct(n)qn =

n≥1

(1 − qnt)t 1 − qn (↑ Olsson ’76) (↓ Granville-Ono ’96)

Positivity. ct(n) > 0 (t ≥ 4).

Monotonicity? ct+1(n) ≥ ct(n)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

SLIDE 22

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47 s-c t-cores arise in rep. thy. of An.

Readable survey by Kleshchev ’10.

Numerical properties: ct(n) = # of t-core partitions of n.

n≥0

ct(n)qn =

n≥1

(1 − qnt)t 1 − qn (↑ Olsson ’76) (↓ Granville-Ono ’96)

Positivity. ct(n) > 0 (t ≥ 4).

Monotonicity? ct+1(n) ≥ ct(n) Modular forms: G.f. for t-cores related to Dedekind’s η-function, a mod. form of wt. 1/2.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

SLIDE 23

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47 s-c t-cores arise in rep. thy. of An.

Readable survey by Kleshchev ’10.

Numerical properties: ct(n) = # of t-core partitions of n.

n≥0

ct(n)qn =

n≥1

(1 − qnt)t 1 − qn (↑ Olsson ’76) (↓ Granville-Ono ’96)

Positivity. ct(n) > 0 (t ≥ 4).

Monotonicity? ct+1(n) ≥ ct(n) Modular forms: G.f. for t-cores related to Dedekind’s η-function, a mod. form of wt. 1/2. Coxeter groups: (↓ Lascoux ’01) t + 1-cores ← → coset reps in At/At

Keys: Bruhat order, Group action!

elements of A A

window notation abacus diagram core partition root lattice point bounded partition reduced expression 4,3,7,10 1,2,1,2

s1s0s2s3s1s0s2s3s1s0

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

SLIDE 24

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47 s-c t-cores arise in rep. thy. of An.

Readable survey by Kleshchev ’10.

Numerical properties: ct(n) = # of t-core partitions of n.

n≥0

ct(n)qn =

n≥1

(1 − qnt)t 1 − qn (↑ Olsson ’76) (↓ Granville-Ono ’96)

Positivity. ct(n) > 0 (t ≥ 4).

Monotonicity? ct+1(n) ≥ ct(n) Modular forms: G.f. for t-cores related to Dedekind’s η-function, a mod. form of wt. 1/2. Coxeter groups: (↓ Lascoux ’01) t + 1-cores ← → coset reps in At/At

Keys: Bruhat order, Group action!

elements of A A

window notation abacus diagram core partition root lattice point bounded partition reduced expression 4,3,7,10 1,2,1,2

s1s0s2s3s1s0s2s3s1s0

elements of C C

window notation abacus diagram core partition root lattice point bounded partition reduced expression

11,9,1,8,16,18 1,2,2

s0s1s0s3s2s1s0s2s3 s2s1s0s2s3s2s1s0

s-c t-cores ← → coset reps in C t/Ct

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

SLIDE 25

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Core partitions in the literature

Representation Theory: (origin) t-cores label t-blocks of irreducible modular representations for Sn. Nakayama cnj. Brauer-Robinson ’47 s-c t-cores arise in rep. thy. of An.

Readable survey by Kleshchev ’10.

Numerical properties: ct(n) = # of t-core partitions of n.

n≥0

ct(n)qn =

n≥1

(1 − qnt)t 1 − qn (↑ Olsson ’76) (↓ Granville-Ono ’96)

Positivity. ct(n) > 0 (t ≥ 4).

Monotonicity? ct+1(n) ≥ ct(n) Modular forms: G.f. for t-cores related to Dedekind’s η-function, a mod. form of wt. 1/2. Coxeter groups: (↓ Lascoux ’01) t + 1-cores ← → coset reps in At/At

Keys: Bruhat order, Group action!

elements of A A

window notation abacus diagram core partition root lattice point bounded partition reduced expression 4,3,7,10 1,2,1,2

s1s0s2s3s1s0s2s3s1s0

elements of C C

window notation abacus diagram core partition root lattice point bounded partition reduced expression

11,9,1,8,16,18 1,2,2

s0s1s0s3s2s1s0s2s3 s2s1s0s2s3s2s1s0

s-c t-cores ← → coset reps in C t/Ct One interpretation: Alcove geometry

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 4 / 15

SLIDE 26

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Type A2 alcoves

s1 s2

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 27

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Type A2 alcoves

s1 s2

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 28

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Type A2 alcoves

s1 s2

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 29

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt. Type A2 alcoves

s1 s2 s0

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 30

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt. Type A2 alcoves

s1 s2 s0

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 31

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves

s1 s2 s0

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 32

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 33

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 34

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Overlay the m-Shi arrangement. Type A2 alcoves

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 35

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Overlay the m-Shi arrangement. Type A2 alcoves

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 36

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Overlay the m-Shi arrangement. Which are representative alcoves? Type A2 alcoves

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 37

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Overlay the m-Shi arrangement. Which are representative alcoves? Type A2 alcoves

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 38

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves Type C2 alcoves Type Ct: generators {s1, . . . , st} Group of signed permutations of {1, . . . , t}. Symmetries of cube or octa’, dim. t.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 39

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves Type C2 alcoves Type Ct: generators {s1, . . . , st} Group of signed permutations of {1, . . . , t}. Symmetries of cube or octa’, dim. t. Add one affine reflection s0 to tile Rt.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 40

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves Type C2 alcoves Type Ct: generators {s1, . . . , st} Group of signed permutations of {1, . . . , t}. Symmetries of cube or octa’, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to s.c. 2t-cores.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 41

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves Type C2 alcoves Type Ct: generators {s1, . . . , st} Group of signed permutations of {1, . . . , t}. Symmetries of cube or octa’, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to s.c. 2t-cores.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 42

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves Type C2 alcoves Type Ct: generators {s1, . . . , st} Group of signed permutations of {1, . . . , t}. Symmetries of cube or octa’, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to s.c. 2t-cores.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 43

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves Type C2 alcoves Type Ct: generators {s1, . . . , st} Group of signed permutations of {1, . . . , t}. Symmetries of cube or octa’, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to s.c. 2t-cores.

Overlay the m-Shi arrangement. Which are representative alcoves?

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 44

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves Type C2 alcoves Type Ct: generators {s1, . . . , st} Group of signed permutations of {1, . . . , t}. Symmetries of cube or octa’, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to s.c. 2t-cores.

Overlay the m-Shi arrangement. Which are representative alcoves?

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 45

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Type A2 alcoves Type C2 alcoves Type Ct: generators {s1, . . . , st} Group of signed permutations of {1, . . . , t}. Symmetries of cube or octa’, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to s.c. 2t-cores.

Overlay the m-Shi arrangement. Which are representative alcoves?

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 46

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcove Geometry

Type At: generators {s1, . . . , st} Group of permutations of {1, . . . , t+1}. Symmetries of regular simplex, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to t +1-cores.

Overlay the m-Shi arrangement. Which are representative alcoves? Type A2 alcoves Type C2 alcoves Type Ct: generators {s1, . . . , st} Group of signed permutations of {1, . . . , t}. Symmetries of cube or octa’, dim. t. Add one affine reflection s0 to tile Rt.

Dom. alcoves correspond to s.c. 2t-cores.

Overlay the m-Shi arrangement. Which are representative alcoves?

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 5 / 15

SLIDE 47

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcoves and simultaneous cores

◮ For all dominant regions in m-Shi arrangement,

the closest alcove to the origin is called m-minimal.

◮ For all bounded dominant regions in m-Shi arrangement,

the furthest alcove from the origin is called m-bounded.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 6 / 15

SLIDE 48

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcoves and simultaneous cores

◮ For all dominant regions in m-Shi arrangement,

the closest alcove to the origin is called m-minimal.

◮ For all bounded dominant regions in m-Shi arrangement,

the furthest alcove from the origin is called m-bounded.

Theorem. (Fishel, Vazirani, ’09–’10)

At alcove is m-minimal ← → corresp. partition is (t, tm + 1)-core. At alcove is m-bounded ← → corresp. partition is (t, tm − 1)-core.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 6 / 15

SLIDE 49

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcoves and simultaneous cores

◮ For all dominant regions in m-Shi arrangement,

the closest alcove to the origin is called m-minimal.

◮ For all bounded dominant regions in m-Shi arrangement,

the furthest alcove from the origin is called m-bounded.

Theorem. (Fishel, Vazirani, ’09–’10)

At alcove is m-minimal ← → corresp. partition is (t, tm + 1)-core. At alcove is m-bounded ← → corresp. partition is (t, tm − 1)-core.

Theorem. (Armstrong, Hanusa, Jones, ’13)

Ct alcove is m-minimal ← → self-conjugate (2t, 2tm + 1)-core. Ct alcove is m-bounded ← → self-conjugate (2t, 2tm − 1)-core.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 6 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Alcoves and simultaneous cores

◮ For all dominant regions in m-Shi arrangement,

the closest alcove to the origin is called m-minimal.

◮ For all bounded dominant regions in m-Shi arrangement,

the furthest alcove from the origin is called m-bounded.

Theorem. (Fishel, Vazirani, ’09–’10)

At alcove is m-minimal ← → corresp. partition is (t, tm + 1)-core. At alcove is m-bounded ← → corresp. partition is (t, tm − 1)-core.

Theorem. (Armstrong, Hanusa, Jones, ’13)

Ct alcove is m-minimal ← → self-conjugate (2t, 2tm + 1)-core. Ct alcove is m-bounded ← → self-conjugate (2t, 2tm − 1)-core. ⋆ Representative alcoves correspond to simultaneous cores. ⋆

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 6 / 15

SLIDE 51

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

The 2-minimal A2 alcoves are (3, 7)-cores

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 7 / 15

SLIDE 52

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-minimal means that when it is reflected closer to to the

rigin, it must pass a hyperplane in the m-Shi arrangement.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 53

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-minimal means that when it is reflected closer to to the

rigin, it must pass a hyperplane in the m-Shi arrangement.

◮ The equivalent abacus interpretation is that

defining bead bi+1 is no more than m levels lower than bi.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 54

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-minimal means that when it is reflected closer to to the

rigin, it must pass a hyperplane in the m-Shi arrangement.

◮ The equivalent abacus interpretation is that

defining bead bi+1 is no more than m levels lower than bi.

◮ Type A: So this t-flush abacus is also (tm + 1)-flush. ◮ At alcove is m-minimal ←

→ (t, tm + 1)-core.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 55

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-minimal means that when it is reflected closer to to the

rigin, it must pass a hyperplane in the m-Shi arrangement.

◮ The equivalent abacus interpretation is that

defining bead bi+1 is no more than m levels lower than bi.

◮ Type A: So this t-flush abacus is also (tm + 1)-flush.

Type C: So this anti-symm. 2t-flush abacus is also (2tm + 1)-flush.

◮ At alcove is m-minimal ←

→ (t, tm + 1)-core. Ct alcove is m-minimal ← → self-conj. (2t, 2tm + 1)-core.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 56

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-minimal means that when it is reflected closer to to the

rigin, it must pass a hyperplane in the m-Shi arrangement.

◮ The equivalent abacus interpretation is that

defining bead bi+1 is no more than m levels lower than bi.

◮ Type A: So this t-flush abacus is also (tm + 1)-flush.

Type C: So this anti-symm. 2t-flush abacus is also (2tm + 1)-flush.

◮ At alcove is m-minimal ←

→ (t, tm + 1)-core. Ct alcove is m-minimal ← → self-conj. (2t, 2tm + 1)-core. Numerical corollary:

◮ dominant At regions ←

→ (t, tm + 1)-cores.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 57

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-minimal means that when it is reflected closer to to the

rigin, it must pass a hyperplane in the m-Shi arrangement.

◮ The equivalent abacus interpretation is that

defining bead bi+1 is no more than m levels lower than bi.

◮ Type A: So this t-flush abacus is also (tm + 1)-flush.

Type C: So this anti-symm. 2t-flush abacus is also (2tm + 1)-flush.

◮ At alcove is m-minimal ←

→ (t, tm + 1)-core. Ct alcove is m-minimal ← → self-conj. (2t, 2tm + 1)-core. Numerical corollary: Agrees with (Athanasiadis, 2004).

◮ dominant At regions ←

→ (t, tm + 1)-cores.

1 t+tm+1

t+tm+1

t

Applications of abacus diagrams:

Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 58

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-minimal means that when it is reflected closer to to the

rigin, it must pass a hyperplane in the m-Shi arrangement.

◮ The equivalent abacus interpretation is that

defining bead bi+1 is no more than m levels lower than bi.

◮ Type A: So this t-flush abacus is also (tm + 1)-flush.

Type C: So this anti-symm. 2t-flush abacus is also (2tm + 1)-flush.

◮ At alcove is m-minimal ←

→ (t, tm + 1)-core. Ct alcove is m-minimal ← → self-conj. (2t, 2tm + 1)-core. Numerical corollary: Agrees with (Athanasiadis, 2004).

◮ dominant At regions ←

→ (t, tm + 1)-cores.

1 t+tm+1

t+tm+1

t

dominant Ct regions ←

→ s-c. (2t, 2tm + 1)-cores. t+tm

t

Applications of abacus diagrams:

Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 59

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-bounded means that when it is reflected further from the

rigin, it must pass a hyperplane in the m-Shi arrangement.

◮ The equivalent abacus interpretation is that

defining bead bi+1 is no more than m levels higher than bi.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 60

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-bounded means that when it is reflected further from the

rigin, it must pass a hyperplane in the m-Shi arrangement.

◮ The equivalent abacus interpretation is that

defining bead bi+1 is no more than m levels higher than bi.

◮ Type A: So this t-flush abacus is also (tm − 1)-flush.

Type C: So this anti-symm. 2t-flush abacus is also (2tm − 1)-flush.

◮ At alcove is m-bounded ←

→ (t, tm − 1)-core. Ct alcove is m-bounded ← → s-c. (2t, 2tm − 1)-core.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 61

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Abaci to the rescue!

Proof sketch:

◮ m-bounded means that when it is reflected further from the

rigin, it must pass a hyperplane in the m-Shi arrangement.

◮ The equivalent abacus interpretation is that

defining bead bi+1 is no more than m levels higher than bi.

◮ Type A: So this t-flush abacus is also (tm − 1)-flush.

Type C: So this anti-symm. 2t-flush abacus is also (2tm − 1)-flush.

◮ At alcove is m-bounded ←

→ (t, tm − 1)-core. Ct alcove is m-bounded ← → s-c. (2t, 2tm − 1)-core. Numerical corollary: Agrees with (Athanasiadis, 2004).

◮ dom. bdd. At regions ←

→ (t, t − 1)-cores.

1 t+tm−1

t+tm−1

t

dom. bdd. Ct regions ←

→ s-c. (2t, 2tm − 1)-cores. t+tm−1

t

Applications of abacus diagrams:

Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 8 / 15

SLIDE 62

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Catalan numbers

Specializing the results of Anderson and Ford, Mai, and Sze, # (t, t + 1)-cores

1 2t+1

2t+1

t

Applications of abacus diagrams:

Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 9 / 15

SLIDE 63

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Catalan numbers

Specializing the results of Anderson and Ford, Mai, and Sze, # (t, t + 1)-cores

1 2t+1

2t+1

t

=

1 t+1

2t

t

Applications of abacus diagrams:

Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 9 / 15

SLIDE 64

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Catalan numbers

Specializing the results of Anderson and Ford, Mai, and Sze, # (t, t + 1)-cores

1 2t+1

2t+1

t

=

1 t+1

2t

t

A Catalan number!

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 9 / 15

SLIDE 65

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Catalan numbers

Specializing the results of Anderson and Ford, Mai, and Sze, # (t, t + 1)-cores

1 2t+1

2t+1

t

=

1 t+1

2t

t

A Catalan number! (of type A)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 9 / 15

SLIDE 66

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Catalan numbers

Specializing the results of Anderson and Ford, Mai, and Sze, # (t, t + 1)-cores

1 2t+1

2t+1

t

=

1 t+1

2t

t

A Catalan number! (of type A)

# self-conj. (2t, 2t + 1)-cores 2t

t

A Catalan number of type C

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 9 / 15

SLIDE 67

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Catalan numbers

Specializing the results of Anderson and Ford, Mai, and Sze, # (t, t + 1)-cores

1 2t+1

2t+1

t

=

1 t+1

2t

t

A Catalan number! (of type A)

# self-conj. (2t, 2t + 1)-cores 2t

t

A Catalan number of type C

Question: Is there a simple statistic on simultaneous core partitions that gives us a q-analog of the Catalan numbers?

λ is a

(t, t + 1)-core

qstat(λ) = 1 [t + 1]q 2t t

q
λ is a self-conj.

(2t, 2t + 1)-core

qstat(λ) = 2t t

q2

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 9 / 15

SLIDE 68

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Catalan numbers

Specializing the results of Anderson and Ford, Mai, and Sze, # (t, t + 1)-cores

1 2t+1

2t+1

t

=

1 t+1

2t

t

A Catalan number! (of type A)

# self-conj. (2t, 2t + 1)-cores 2t

t

A Catalan number of type C

Question: Is there a simple statistic on simultaneous core partitions that gives us a q-analog of the Catalan numbers?

λ is a

(t, t + 1)-core

qstat(λ) = 1 [t + 1]q 2t t

q
λ is a self-conj.

(2t, 2t + 1)-core

qstat(λ) = 2t t

q2

Answer: Yes. We will create an analog of the major statistic.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 9 / 15

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Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

The major statistic

Given a permutation π of {1, . . . , n} written in one-line notation as π = π1π2 · · · πn, the major statistic maj(π) is defined as the sum

f the positions of the descents of π, in other words,

maj(π) =

i:πi−1>πi

i.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 10 / 15

SLIDE 70

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

The major statistic

Given a permutation π of {1, . . . , n} written in one-line notation as π = π1π2 · · · πn, the major statistic maj(π) is defined as the sum

f the positions of the descents of π, in other words,

maj(π) =

i:πi−1>πi

i. Named in honor of Major Percy MacMahon who showed it has the same distribution as the statistic of the number of inversions:

π∈Sn

qmaj(π) =

π∈Sn

qinv(π)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 10 / 15

SLIDE 71

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

A major statistic for simultaneous cores

Let λ be a (t, t + 1)-core. Define b = (b0, . . . , bt−1) where bi = # 1st col. boxes with hook length ≡ i mod t.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 11 / 15

SLIDE 72

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

A major statistic for simultaneous cores

Let λ be a (t, t + 1)-core. Define b = (b0, . . . , bt−1) where bi = # 1st col. boxes with hook length ≡ i mod t. Define maj(λ) =

i : bi−1≥bi

(2i − bi).

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 11 / 15

SLIDE 73

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

A major statistic for simultaneous cores

Let λ be a (t, t + 1)-core. Define b = (b0, . . . , bt−1) where bi = # 1st col. boxes with hook length ≡ i mod t. Define maj(λ) =

i : bi−1≥bi

(2i − bi).

Theorem. (AHJ ’13)
λ is a

(t, t + 1)-core

qmaj(λ) = 1 [t + 1]q 2t t

q

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 11 / 15

SLIDE 74

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

A major statistic for simultaneous cores

Let λ be a (t, t + 1)-core. Define b = (b0, . . . , bt−1) where bi = # 1st col. boxes with hook length ≡ i mod t. Define maj(λ) =

i : bi−1≥bi

(2i − bi).

Theorem. (AHJ ’13)
λ is a

(t, t + 1)-core

qmaj(λ) = 1 [t + 1]q 2t t

q

Let λ be a s-c. (2t, 2t + 1)-core. Define b = (b0, . . . , bt) where b0 = 0 and bi = (# diag. arms ≡ i mod 2t) − (# diag. arms ≡ 2t−i+1 mod 2t)

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 11 / 15

SLIDE 75

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

A major statistic for simultaneous cores

Let λ be a (t, t + 1)-core. Define b = (b0, . . . , bt−1) where bi = # 1st col. boxes with hook length ≡ i mod t. Define maj(λ) =

i : bi−1≥bi

(2i − bi).

Theorem. (AHJ ’13)
λ is a

(t, t + 1)-core

qmaj(λ) = 1 [t + 1]q 2t t

q

Let λ be a s-c. (2t, 2t + 1)-core. Define b = (b0, . . . , bt) where b0 = 0 and bi = (# diag. arms ≡ i mod 2t) − (# diag. arms ≡ 2t−i+1 mod 2t) Define maj(λ) = 2

i : bi−1≥bi

(2i − bi − 1).

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 11 / 15

SLIDE 76

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

A major statistic for simultaneous cores

Let λ be a (t, t + 1)-core. Define b = (b0, . . . , bt−1) where bi = # 1st col. boxes with hook length ≡ i mod t. Define maj(λ) =

i : bi−1≥bi

(2i − bi).

Theorem. (AHJ ’13)
λ is a

(t, t + 1)-core

qmaj(λ) = 1 [t + 1]q 2t t

q

Let λ be a s-c. (2t, 2t + 1)-core. Define b = (b0, . . . , bt) where b0 = 0 and bi = (# diag. arms ≡ i mod 2t) − (# diag. arms ≡ 2t−i+1 mod 2t) Define maj(λ) = 2

i : bi−1≥bi

(2i − bi − 1).

Theorem. (AHJ ’13)
λ is a self-conj.

(2t, 2t + 1)-core

qmaj(λ) = 2t t

q2

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 11 / 15

SLIDE 77

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

A major statistic for simultaneous cores

Let λ be a (t, t + 1)-core. Define b = (b0, . . . , bt−1) where bi = # 1st col. boxes with hook length ≡ i mod t. Define maj(λ) =

i : bi−1≥bi

(2i − bi).

Theorem. (AHJ ’13)
λ is a

(t, t + 1)-core

qmaj(λ) = 1 [t + 1]q 2t t

q

Let λ be a s-c. (2t, 2t + 1)-core. Define b = (b0, . . . , bt) where b0 = 0 and bi = (# diag. arms ≡ i mod 2t) − (# diag. arms ≡ 2t−i+1 mod 2t) Define maj(λ) = 2

i : bi−1≥bi

(2i − bi − 1).

Theorem. (AHJ ’13)
λ is a self-conj.

(2t, 2t + 1)-core

qmaj(λ) = 2t t

q2

Note: maj defined as a sum

ver descents in a sequence.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 11 / 15

SLIDE 78

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

A major statistic for abacus diagrams

Let λ be a (t, t + 1)-core. Read off the levels of the defining beads of the (normalized) abacus to give b = (b0, . . . , bt−1). Define maj(λ) =

i : bi−1≥bi

(2i − bi). Then

λ is a

(t, t + 1)-core

qmaj(λ) = 1 [t + 1]q 2t t

q

Let λ be a s-c. (2t, 2t + 1)-core. Read off the levels of the defining beads of the corresponding abacus to give b = (b0, . . . , bt). Define maj(λ) = 2

i : bi−1≥bi

(2i − bi − 1). Then

λ is a self-conj.

(2t, 2t + 1)-core

qmaj(λ) = 2t t

q2

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 11 / 15

SLIDE 79

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Proof sketch

◮ Use Anderson’s lattice path bijection:

(s, t)-flush abaci ← → L : (0, 0) → (s, t) above y = t

s x.

−4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 −4 −8 −12 −16 −20 −24 −28 −32 −36 −40 −44 −48 −52 9 5 1 −3 −7 −11 −15 −19 −23 −27 −31 −35 −39 22 18 14 10 6 2 −2 −6 −10 −14 −18 −22 −26 35 31 27 23 19 15 11 7 3 −1 −5 −9 −13

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 12 / 15

SLIDE 80

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Proof sketch

◮ Use Anderson’s lattice path bijection:

(s, t)-flush abaci ← → L : (0, 0) → (s, t) above y = t

s x.

−4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 −4 −8 −12 −16 −20 −24 −28 −32 −36 −40 −44 −48 −52 9 5 1 −3 −7 −11 −15 −19 −23 −27 −31 −35 −39 22 18 14 10 6 2 −2 −6 −10 −14 −18 −22 −26 35 31 27 23 19 15 11 7 3 −1 −5 −9 −13

◮ Create a similar lattice path bijection: (improves Ford-Mai-Sze)

antisymm. (s, t)-flush abaci ←

→ L : (0, 0) → s

2

,

t

2

.

−23 −22 −21 −20 −19 −18 −17 −16 −15 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

3 −5 −13 −21 −29 −37 −45 −53 −61 −69 −77 −85 −93 16 8 −8 −16 −24 −32 −40 −48 −56 −64 −72 −80 29 21 13 5 −3 −11 −19 −27 −35 −43 −51 −59 −67 42 34 26 18 10 2 −6 −14 −22 −30 −38 −46 −54 55 47 39 31 23 15 7 −1 −9 −17 −25 −33 −41 68 60 52 44 36 28 20 12 4 −4 −12 −20 −28 81 73 65 57 49 41 33 25 17 9 1 −7 −15 94 86 78 70 62 54 46 38 30 22 14 6 −2

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 12 / 15

SLIDE 81

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Proof sketch

◮ (t, t + 1)-flush abaci ←

→ L : (0, 0) → (t, t) above y = x. Dyck paths!

−3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 12 / 15

SLIDE 82

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Proof sketch

◮ (t, t + 1)-flush abaci ←

→ L : (0, 0) → (t, t) above y = x. Dyck paths!

−3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1

◮ antisymm. (2t, 2t + 1)-flush abaci ←

→ L : (0, 0) → (t, t).

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 12 / 15

SLIDE 83

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Proof sketch

◮ (t, t + 1)-flush abaci ←

→ L : (0, 0) → (t, t) above y = x. Dyck paths!

−3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1

◮ antisymm. (2t, 2t + 1)-flush abaci ←

→ L : (0, 0) → (t, t).

◮ Use the major index on lattice paths that is known to give the

desired q-analog: maj(L) =

i:(Li,Li+1)=(E,N)

i

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 12 / 15

SLIDE 84

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Proof sketch

◮ (t, t + 1)-flush abaci ←

→ L : (0, 0) → (t, t) above y = x. Dyck paths!

−3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1

◮ antisymm. (2t, 2t + 1)-flush abaci ←

→ L : (0, 0) → (t, t).

◮ Use the major index on lattice paths that is known to give the

desired q-analog: maj(L) =

i:(Li,Li+1)=(E,N)

i q0 + q2 + q3 + q4 + q2+4 =

1 [4]q

6

3

q

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 12 / 15

SLIDE 85

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Proof sketch

◮ (t, t + 1)-flush abaci ←

→ L : (0, 0) → (t, t) above y = x. Dyck paths!

−3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1

◮ antisymm. (2t, 2t + 1)-flush abaci ←

→ L : (0, 0) → (t, t).

◮ Use the major index on lattice paths that is known to give the

desired q-analog: maj(L) =

i:(Li,Li+1)=(E,N)

i q0 + q2 + q3 + q4 + q2+4 =

1 [4]q

6

3

q

q0 + q1 + q2 + q2 + q3 + q1+3 = 4

2

q

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 12 / 15

SLIDE 86

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Proof sketch

◮ (t, t + 1)-flush abaci ←

→ L : (0, 0) → (t, t) above y = x. Dyck paths!

−3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1 −3 −6 −9 1 −2 −5 5 2 −1

◮ antisymm. (2t, 2t + 1)-flush abaci ←

→ L : (0, 0) → (t, t).

◮ Use the major index on lattice paths that is known to give the

desired q-analog: maj(L) =

i:(Li,Li+1)=(E,N)

i q0 + q2 + q3 + q4 + q2+4 =

1 [4]q

6

3

q

q0 + q1 + q2 + q2 + q3 + q1+3 = 4

2

q

◮ Translate this major index to language of abaci and cores.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 12 / 15

SLIDE 87

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Talk Recap

◮ Definitions

◮ Core partitions and abacus diagrams ◮ Simultaneity Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 13 / 15

SLIDE 88

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Talk Recap

◮ Definitions

◮ Core partitions and abacus diagrams ◮ Simultaneity

◮ Alcove geometry

◮ Which alcoves are good representatives? ◮ Simultaneous core partitions! Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 13 / 15

SLIDE 89

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Talk Recap

◮ Definitions

◮ Core partitions and abacus diagrams ◮ Simultaneity

◮ Alcove geometry

◮ Which alcoves are good representatives? ◮ Simultaneous core partitions!

◮ Search for q-analogs of Catalan numbers

◮ Piggy-back on lattice path combinatorics ◮ A new major statistic on simultaneous cores. Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 13 / 15

SLIDE 90

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Talk Recap

◮ Definitions

◮ Core partitions and abacus diagrams ◮ Simultaneity

◮ Alcove geometry

◮ Which alcoves are good representatives? ◮ Simultaneous core partitions!

◮ Search for q-analogs of Catalan numbers

◮ Piggy-back on lattice path combinatorics ◮ A new major statistic on simultaneous cores.

◮ Remarkable

◮ Type-independent setup. ◮ Abaci are the right tool. Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 13 / 15

SLIDE 91

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

What’s next?

1. Core survey

◮ Compile combinatorial interpretations into illustrated dictionary. ◮ Reconcile many appearances of cores into historical survey. ◮ Gathering sources stage — What do you know?

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 14 / 15

SLIDE 92

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

What’s next?

1. Core survey
2. Open question: Catalan q-analogs

◮ Question. Is there a core statistic for a q-analog of 1 a+b

a+b

a

?

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 14 / 15

SLIDE 93

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

What’s next?

1. Core survey
2. Open question: Catalan q-analogs

◮ Question. Is there a core statistic for a q-analog of 1 a+b

a+b

a

?

◮ Question. Is there a core statistic for m-Catalan (t, tm ± 1)?

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 14 / 15

SLIDE 94

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

What’s next?

1. Core survey
2. Open question: Catalan q-analogs

◮ Question. Is there a core statistic for a q-analog of 1 a+b

a+b

a

?

◮ Question. Is there a core statistic for m-Catalan (t, tm ± 1)? ◮ Progress: m-Catalan number C3 through (3, 3m + 1)-cores.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 14 / 15

SLIDE 95

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

What’s next?

1. Core survey
2. Open question: Catalan q-analogs

◮ Question. Is there a core statistic for a q-analog of 1 a+b

a+b

a

?

◮ Question. Is there a core statistic for m-Catalan (t, tm ± 1)? ◮ Progress: m-Catalan number C3 through (3, 3m + 1)-cores. ◮ Progress: Drew has a candidate statistic.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 14 / 15

SLIDE 96

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

What’s next?

1. Core survey
2. Open question: Catalan q-analogs

◮ Question. Is there a core statistic for m-Catalan (t, tm ± 1)? ◮ Progress: m-Catalan number C3 through (3, 3m + 1)-cores.

3. Open question: Properties of simultaneous cores

◮ Question. What is the average size of an (s, t)-core partition? ◮ Progress: Answer:

Proof?

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 14 / 15

SLIDE 97

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

What’s next?

1. Core survey
2. Open question: Catalan q-analogs

◮ Question. Is there a core statistic for m-Catalan (t, tm ± 1)? ◮ Progress: m-Catalan number C3 through (3, 3m + 1)-cores.

3. Open question: Properties of simultaneous cores

◮ Question. What is the average size of an (s, t)-core partition? ◮ Progress: Answer: (s + t + 1)(s − 1)(t − 1)/24. Proof?

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 14 / 15

SLIDE 98

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

What’s next?

1. Core survey
2. Open question: Catalan q-analogs

◮ Question. Is there a core statistic for m-Catalan (t, tm ± 1)? ◮ Progress: m-Catalan number C3 through (3, 3m + 1)-cores.

3. Open question: Properties of simultaneous cores

◮ Question. What is the average size of an (s, t)-core partition? ◮ Progress: Answer: (s + t + 1)(s − 1)(t − 1)/24. Proof?

4. Open question: Cyclic sieving phenomenon

◮ Note: 1 [a+b]q

a+b

a

q
q=−1 =

⌊ a

2⌋+⌊ b 2 ⌋

⌊ a

2 ⌋

.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 14 / 15

SLIDE 99

Core partitions & Abacus diagrams Alcoves q-Catalan numbers Coming attractions

Thank you!

Slides available: people.qc.cuny.edu/chanusa > Talks Interact: people.qc.cuny.edu/chanusa > Animations Drew Armstrong, Christopher R. H. Hanusa, Brant C. Jones. Results and conjectures on simultaneous core partitions. Submitted, 2013. arXiv:1308.0572. Christopher R. H. Hanusa and Brant C. Jones. Abacus models for parabolic quotients of affine Coxeter groups Journal of Algebra. Vol. 361, 134–162. (2012) arXiv:1105.5333 Gordon James and Adalbert Kerber. The representation theory of the symmetric group. Addison-Wesley, 1981.

Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic Christopher R. H. Hanusa Queens College, CUNY November 4, 2013 15 / 15