Simultaneous Core Partitions for Affine Weyl Groups

Simultaneous core partitions for aﬃne Weyl groups

Marko Thiel

Universit¨atZ¨urich

Joint work with Nathan Williams (LACIM, Montr´eal)

Marko Thiel Simultaneous core partitions for affine Weyl groups – Sea-action on a-cores – The geometry of the affine symmetric group Sea – The lattice of cores – Simultaneous (a, b)-cores – Other affine Weyl groups

– a-core partitions

Outline

Marko Thiel Simultaneous core partitions for affine Weyl groups – The geometry of the affine symmetric group Sea – The lattice of cores – Simultaneous (a, b)-cores – Other affine Weyl groups

– Sea-action on a-cores

Outline – a-core partitions

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups – The lattice of cores – Simultaneous (a, b)-cores – Other aﬃne Weyl groups

– The geometry of the aﬃne symmetric group Sea

Outline – a-core partitions

– Sea-action on a-cores

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups – Simultaneous (a, b)-cores – Other aﬃne Weyl groups

– The lattice of cores

Outline – a-core partitions

– Sea-action on a-cores – The geometry of the aﬃne symmetric group Sea

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups – Other aﬃne Weyl groups

– Simultaneous (a, b)-cores

Outline – a-core partitions

– Sea-action on a-cores – The geometry of the aﬃne symmetric group Sea – The lattice of cores

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups – Other aﬃne Weyl groups

Outline – a-core partitions

– Sea-action on a-cores – The geometry of the aﬃne symmetric group Sea – The lattice of cores – Simultaneous (a, b)-cores

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Outline – a-core partitions

– Sea-action on a-cores – The geometry of the aﬃne symmetric group Sea – The lattice of cores – Simultaneous (a, b)-cores – Other aﬃne Weyl groups

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6, 4, 2, 1, 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 { } { }

Integer partitions

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Example: λ = (6, 4, 2, 1, 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 { } { }

Integer partitions

An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 { } { }

Integer partitions

An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6, 4, 2, 1, 1) is an integer partition of size 14.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 { } { }

Integer partitions

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Hook length of a box: # boxes below it + # boxes to the right of it + 1 { } { }

Integer partitions

An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6, 4, 2, 1, 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Integer partitions

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups An a-core is an integer partition with no hook length equal to a. Example: λ = (6, 4, 2, 1, 1) is a 3-core. 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic.

a-core partitions

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Example: λ = (6, 4, 2, 1, 1) is a 3-core. 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic.

a-core partitions

An a-core is an integer partition with no hook length equal to a.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic.

a-core partitions

An a-core is an integer partition with no hook length equal to a. Example: λ = (6, 4, 2, 1, 1) is a 3-core.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic.

a-core partitions

An a-core is an integer partition with no hook length equal to a. Example: λ = (6, 4, 2, 1, 1) is a 3-core. 107 5 4 2 1 8 4 2 1 4 1 2 1

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups a-core partitions

An a-core is an integer partition with no hook length equal to a. Example: λ = (6, 4, 2, 1, 1) is a 3-core. 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The a-residue of a box in row i and column j is r(i, j) = j i mod a. − 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 removable box: Can remove this box and still have a Ferrers shape.

Adding/Removing boxes

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 removable box: Can remove this box and still have a Ferrers shape.

Adding/Removing boxes

The a-residue of a box in row i and column j is r(i, j) = j i mod a. −

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups removable box: Can remove this box and still have a Ferrers shape.

Adding/Removing boxes

The a-residue of a box in row i and column j is r(i, j) = j i mod a. − 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Adding/Removing boxes

The a-residue of a box in row i and column j is r(i, j) = j i mod a. − 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 removable box: Can remove this box and still have a Ferrers shape.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups addable box: Can add this box and still have a Ferrers shape.

Adding/Removing boxes

The a-residue of a box in row i and column j is r(i, j) = j i mod a. − 0 1 2 0 1 2 2 0 1 2 1 2 0 2 removable box: Can remove this box and still have a Ferrers shape.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Adding/Removing boxes

The a-residue of a box in row i and column j is r(i, j) = j i mod a. − 0 1 2 0 1 2 2 0 1 2 1 2 0 2 removable box: Can remove this box and still have a Ferrers shape. addable box: Can add this box and still have a Ferrers shape.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Note: There are never both addable and removable boxes of the same residue.

Adding/Removing boxes

The a-residue of a box in row i and column j is r(i, j) = j i mod a. − 0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 0 1 2 1 removable box: Can remove this box and still have a Ferrers shape. addable box: Can add this box and still have a Ferrers shape.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Adding/Removing boxes

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 2 – si = e – si sj = sj si if j / i + 1, i 1 ∈ { − } – si si+1si = si+1si si+1

Let si act on an a-core by either adding all addable boxes of residue i or removing all removable boxes of residue i. The result is again an a-core. Relations:

These are the Coxeter relations of the aﬃne symmetric group Sea!

Sea = s0, s1,..., sa−1 Relations h | i

Action of Sea

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups – si sj = sj si if j / i + 1, i 1 ∈ { − } – si si+1si = si+1si si+1

2 – si = e

The result is again an a-core. Relations:

These are the Coxeter relations of the aﬃne symmetric group Sea!

Sea = s0, s1,..., sa−1 Relations h | i

Action of Sea

Let si act on an a-core by either adding all addable boxes of residue i or removing all removable boxes of residue i.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups – si si+1si = si+1si si+1

2 – si = e – si sj = sj si if j / i + 1, i 1 ∈ { − }

Relations:

These are the Coxeter relations of the aﬃne symmetric group Sea!

Sea = s0, s1,..., sa−1 Relations h | i

Action of Sea

Let si act on an a-core by either adding all addable boxes of residue i or removing all removable boxes of residue i. The result is again an a-core.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups – si sj = sj si if j / i + 1, i 1 ∈ { − } – si si+1si = si+1si si+1 These are the Coxeter relations of the aﬃne symmetric group Sea!

Sea = s0, s1,..., sa−1 Relations h | i

Action of Sea

Let si act on an a-core by either adding all addable boxes of residue i or removing all removable boxes of residue i. The result is again an a-core. Relations: 2 – si = e

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups – si si+1si = si+1si si+1 These are the Coxeter relations of the aﬃne symmetric group Sea!

Sea = s0, s1,..., sa−1 Relations h | i

Action of Sea

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups These are the Coxeter relations of the aﬃne symmetric group Sea!

Sea = s0, s1,..., sa−1 Relations h | i

Action of Sea

Let si act on an a-core by either adding all addable boxes of residue i or removing all removable boxes of residue i. The result is again an a-core. Relations: 2 – si = e – si sj = sj si if j / i + 1, i 1 ∈ { − } – si si+1si = si+1si si+1

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups These are the Coxeter relations of the aﬃne symmetric group Sea!

Sea = s0, s1,..., sa−1 Relations h | i

Action of Sea

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Sea = s0, s1,..., sa−1 Relations h | i

Action of Sea

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Sea = s0, s1,..., sa−1 Relations h | i

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

∅

Action of Sea

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups ∅

Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

s0 0 ∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

s0 s1 0 0 1 ∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

s0 s1 s2 0 0 1 0 1 2 ∅ 2

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

s0 s1 s2 s0 0 0 1 0 1 2 0 1 2 0 ∅ 2 2 0

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

s0 s1 s2 s0 0 0 1 0 1 2 0 1 2 0 ∅ 2 2 0

s1 0 1 2 0 1 2 0 1 1

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

s0 s1 s2 s0 0 0 1 0 1 2 0 1 2 0 ∅ 2 2 0

s1 0 1 2 0 1 s0 0 1 2 0 1 2 0 1 2 0 1 1 1 0

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

s0 s1 s2 s0 0 0 1 0 1 2 0 1 2 0 ∅ 2 2 0

s1 0 1 2 0 1 s0 0 1 2 0 1 s2 0 1 2 0 1 2 2 0 1 2 0 1 2 0 1 2 1 1 1 2 0 0 2

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Action of Sea

Example: Calculate s2s0s1s0s2s1s0 (a=3): · ∅

s0 s1 s2 s0 0 0 1 0 1 2 0 1 2 0 ∅ 2 2 0

s1 0 1 2 0 1 s0 0 1 2 0 1 s2 0 1 2 0 1 2 2 0 1 2 0 1 2 0 1 2 1 1 1 2 0 0 2 Can get all a-cores by acting on the empty a-core by some ∅ element we of Sea (transitive action).

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. −

2 0 A ◦ 1

The aﬃne reﬂection group Sea

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Let si act by reﬂection through the wall labelled i.

2 0 A ◦ 1

The aﬃne reﬂection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. −

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 2 0 A ◦ 1

The aﬃne reﬂection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. −

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. −

2 0 A ◦ 1

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. −

s0A ◦ 2 2 0 1 A ◦ 1

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. −

s0A ◦ 2 2 0 1 2 A ◦ 1 0

s1s0A ◦

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. −

s s s A s0A 2 1 0 ◦ ◦ 2 1 2 0 1 2 0 A ◦ 1 0

s1s0A ◦

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. −

s s s A s0A 2 1 0 ◦ ◦ 2 1 2 0 1 2 0 1 A ◦ 1 0 2

s1s0A ◦ s0s2s1s0A ◦

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. −

s s s A s0A 2 1 0 ◦ ◦ 2 1 0 2 0 1 2 0 1 2 s1s0s2s1s0A A ◦ ◦ 1 0 2

s1s0A ◦ s0s2s1s0A ◦

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. −

s0s1s0s2s1s0A ◦ s s s A s0A 2 1 0 ◦ ◦ 1 2 2 1 0 2 0 1 2 0 1 2 s1s0s2s1s0A A ◦ ◦ 1 0 2

s1s0A ◦ s0s2s1s0A ◦

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea

Have a special simplex (alcove) A◦ with walls labelled by 0, 1,..., a 1. Let si act by reﬂection through the wall labelled i. − s2s0s1s0s2s1s0A ◦ s0s1s0s2s1s0A ◦ 1 s s s A s0A 2 1 0 ◦ ◦ 1 2 0 2 1 0 2 0 1 2 0 1 2 s1s0s2s1s0A A ◦ ◦ 1 0 2

s1s0A ◦ s0s2s1s0A ◦

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 0 2 1 0 0 1 2 0 1 2 0 1 2 0 2 1 0 2 1 2 0 1 2 0 1 2 0 1 2 0 1 1 0 2 1 0 2 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 2 1 0 2 1 0 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 2 1 0 2 1 0 2 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 0 2 1 0 2 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 0 2 1 0 2 2 0 1 2 0 1 2 0 1 2 0 1 2 1 0 2 1 0 1 2 0 1 2 0 1 2 0 0 2 1 0

The aﬃne reﬂection group Sea

Action by all of Sea on A◦ gives every alcove exactly once (simply transitive action).

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea

Action by all of Sea on A◦ gives every alcove exactly once (simply transitive action). 0 2 1 0 0 1 2 0 1 2 0 1 2 0 2 1 0 2 1 2 0 1 2 0 1 2 0 1 2 0 1 1 0 2 1 0 2 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 2 1 0 2 1 0 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 2 1 0 2 1 0 2 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 0 2 1 0 2 1 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 0 2 1 0 2 2 0 1 2 0 1 2 0 1 2 0 1 2 1 0 2 1 0 1 2 0 1 2 0 1 2 0 0 2 1 0

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups wA◦ w e 7→ e · ∅

∅ ∅ ∅

The aﬃne reﬂection group Sea acting on a-cores

View the action of Sea on a-cores geometrically:

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups ∅ ∅ ∅

∅ ∅ ∅

The aﬃne reﬂection group Sea acting on a-cores

View the action of Sea on a-cores geometrically:

wA◦ w e 7→ e · ∅

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea acting on a-cores

View the action of Sea on a-cores geometrically:

wA◦ w e 7→ e · ∅

∅ ∅ ∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups In every chamber, have every a-core exactly once.

The aﬃne reﬂection group Sea acting on a-cores

∅ ∅ ∅

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea acting on a-cores

∅ ∅ ∅

In every chamber, have every a-core exactly once.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Bijection: Dominant alcoves a-cores ↔

∅

The aﬃne reﬂection group Sea acting on a-cores

Call one chamber dominant.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups ∅

The aﬃne reﬂection group Sea acting on a-cores

Call one chamber dominant. Bijection: Dominant alcoves a-cores ↔

Marko Thiel Simultaneous core partitions for affine Weyl groups The affine reflection group Sea acting on a-cores

Call one chamber dominant. Bijection: Dominant alcoves a-cores ↔

∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups ∅

The lattice of cores

Invert the picture! −1 wA◦ w A◦ e 7→ e

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The lattice of cores

Invert the picture! −1 wA◦ w A◦ e 7→ e

∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The lattice of cores

The coroot lattice Qˇ is the lattice of cores. Bijection:

core ˇ : a-cores Qˇ Q { } →

∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups ρ a

The size of cores

For an a-core λ and µ = coreQˇ(λ), we have (ρ is the Weyl vector) a ρ 2 ρ 2 size(λ) = µ 2 − a − a

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The size of cores

For an a-core λ and µ = coreQˇ(λ), we have (ρ is the Weyl vector) a ρ 2 ρ 2 size(λ) = µ 2 − a − a

ρ a

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups a ρ 2 ρ 2 X size(λ) X µ− − q = q 2 k a k k a k λ a-core µ∈Qˇ ∞ Y (1 qai )a = − 1 qi i=1 − using a formula of Macdonald. The classical proof of this is more direct.

The size of cores

So we get the generating function that counts a-cores by size:

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 2 2 X a µ− ρ − ρ = q 2 k a k k a k µ∈Qˇ ∞ Y (1 qai )a = − 1 qi i=1 − using a formula of Macdonald. The classical proof of this is more direct.

The size of cores

So we get the generating function that counts a-cores by size: X qsize(λ) λ a-core

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups ∞ Y (1 qai )a = − 1 qi i=1 − using a formula of Macdonald. The classical proof of this is more direct.

The size of cores

So we get the generating function that counts a-cores by size:

a ρ 2 ρ 2 X size(λ) X µ− − q = q 2 k a k k a k λ a-core µ∈Qˇ

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups using a formula of Macdonald. The classical proof of this is more direct.

The size of cores

So we get the generating function that counts a-cores by size:

a ρ 2 ρ 2 X size(λ) X µ− − q = q 2 k a k k a k λ a-core µ∈Qˇ ∞ Y (1 qai )a = − 1 qi i=1 −

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The classical proof of this is more direct.

The size of cores

So we get the generating function that counts a-cores by size:

a ρ 2 ρ 2 X size(λ) X µ− − q = q 2 k a k k a k λ a-core µ∈Qˇ ∞ Y (1 qai )a = − 1 qi i=1 − using a formula of Macdonald.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The size of cores

So we get the generating function that counts a-cores by size:

a ρ 2 ρ 2 X size(λ) X µ− − q = q 2 k a k k a k λ a-core µ∈Qˇ ∞ Y (1 qai )a = − 1 qi i=1 − using a formula of Macdonald. The classical proof of this is more direct.

Marko Thiel Simultaneous core partitions for affine Weyl groups A simultaneous( a, b)-core is an a-core that is also a b-core. That is, an (a, b)-core is an integer partition with no hooklength equal to either a or b. There are finitely many( a, b)-cores if and only if a and b are relatively prime. So we restrict to that case. Our first aim is to count the number of (a, b)-cores.

Simultaneous core partitions

Marko Thiel Simultaneous core partitions for affine Weyl groups That is, an (a, b)-core is an integer partition with no hooklength equal to either a or b. There are finitely many( a, b)-cores if and only if a and b are relatively prime. So we restrict to that case. Our first aim is to count the number of (a, b)-cores.

Simultaneous core partitions

A simultaneous( a, b)-core is an a-core that is also a b-core.

Marko Thiel Simultaneous core partitions for affine Weyl groups There are finitely many( a, b)-cores if and only if a and b are relatively prime. So we restrict to that case. Our first aim is to count the number of (a, b)-cores.

Simultaneous core partitions

A simultaneous( a, b)-core is an a-core that is also a b-core. That is, an (a, b)-core is an integer partition with no hooklength equal to either a or b.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups So we restrict to that case. Our ﬁrst aim is to count the number of (a, b)-cores.

Simultaneous core partitions

A simultaneous( a, b)-core is an a-core that is also a b-core. That is, an (a, b)-core is an integer partition with no hooklength equal to either a or b. There are ﬁnitely many( a, b)-cores if and only if a and b are relatively prime.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Our ﬁrst aim is to count the number of (a, b)-cores.

Simultaneous core partitions

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Simultaneous core partitions

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups ∅

Simultaneous core partitions

The simultaneous (3, 4)-cores in green:

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Simultaneous core partitions

The simultaneous (3, 4)-cores in green:

∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups (a, b)-cores Qˇ a(b). So we want to count Qˇ a(b). ↔ ∩ S ∩ S

Simultaneous core partitions

They are the lattice points in a simplex 3(4)! S

∅

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups So we want to count Qˇ a(b). ∩ S

Simultaneous core partitions

They are the lattice points in a simplex 3(4)! S

∅

(a, b)-cores Qˇ a(b). ↔ ∩ S

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Simultaneous core partitions

They are the lattice points in a simplex 3(4)! S

∅

(a, b)-cores Qˇ a(b). So we want to count Qˇ a(b). ↔ ∩ S ∩ S

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 1 If V is an n-dimensional vector space, L is a lattice and P is a polytope with vertices in L, then for positive integers b

G(b) := #(L bP) ∩ is a polynomial of degree n in b. 2 Furthermore, we have

G( b) = ( 1)n#(L bP◦). − − ∩ −

Ehrhart theory

Theorem (Ehrhart)

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 2 Furthermore, we have

G( b) = ( 1)n#(L bP◦). − − ∩ −

Ehrhart theory

Theorem (Ehrhart) 1 If V is an n-dimensional vector space, L is a lattice and P is a polytope with vertices in L, then for positive integers b

G(b) := #(L bP) ∩ is a polynomial of degree n in b.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Ehrhart theory

Theorem (Ehrhart) 1 If V is an n-dimensional vector space, L is a lattice and P is a polytope with vertices in L, then for positive integers b

G(b) := #(L bP) ∩ is a polynomial of degree n in b. 2 Furthermore, we have

G( b) = ( 1)n#(L bP◦). − − ∩ −

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 1 a(b) is not the b-th dilation of a polytope. Solution: it is S congruent to bA◦! 2 The vertices of A◦ are not in Qˇ. Solution: omitted.

Conclusion: The number of (a, b)-cores is a polynomial of degree a 1 in b. It− is determined uniquely by its evaluation at a points. Result: The number of (a, b)-cores is the rational Catalan number 1 a + b Cat(a, b) := . a + b a The classical (combinatorial) proof is much simpler. Next goal: average size of an (a, b)-core.

Ehrhart theory

Two problems with applying this to count Qˇ a(b): ∩ S

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 2 The vertices of A◦ are not in Qˇ. Solution: omitted. Conclusion: The number of (a, b)-cores is a polynomial of degree a 1 in b. It− is determined uniquely by its evaluation at a points. Result: The number of (a, b)-cores is the rational Catalan number 1 a + b Cat(a, b) := . a + b a The classical (combinatorial) proof is much simpler. Next goal: average size of an (a, b)-core.

Ehrhart theory

Two problems with applying this to count Qˇ a(b): ∩ S 1 a(b) is not the b-th dilation of a polytope. Solution: it is S congruent to bA◦!

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Conclusion: The number of (a, b)-cores is a polynomial of degree a 1 in b. It− is determined uniquely by its evaluation at a points. Result: The number of (a, b)-cores is the rational Catalan number 1 a + b Cat(a, b) := . a + b a The classical (combinatorial) proof is much simpler. Next goal: average size of an (a, b)-core.

Ehrhart theory

Two problems with applying this to count Qˇ a(b): ∩ S 1 a(b) is not the b-th dilation of a polytope. Solution: it is S congruent to bA◦! 2 The vertices of A◦ are not in Qˇ. Solution: omitted.

Ehrhart theory

Two problems with applying this to count Qˇ a(b): ∩ S 1 a(b) is not the b-th dilation of a polytope. Solution: it is S congruent to bA◦! 2 The vertices of A◦ are not in Qˇ. Solution: omitted.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups It is determined uniquely by its evaluation at a points. Result: The number of (a, b)-cores is the rational Catalan number 1 a + b Cat(a, b) := . a + b a The classical (combinatorial) proof is much simpler. Next goal: average size of an (a, b)-core.

Ehrhart theory

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Result: The number of (a, b)-cores is the rational Catalan number 1 a + b Cat(a, b) := . a + b a The classical (combinatorial) proof is much simpler. Next goal: average size of an (a, b)-core.

Ehrhart theory

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The classical (combinatorial) proof is much simpler. Next goal: average size of an (a, b)-core.

Ehrhart theory

Two problems with applying this to count Qˇ a(b): ∩ S 1 a(b) is not the b-th dilation of a polytope. Solution: it is S congruent to bA◦! 2 The vertices of A◦ are not in Qˇ. Solution: omitted. Conclusion: The number of (a, b)-cores is a polynomial of degree a 1 in b. It− is determined uniquely by its evaluation at a points. Result: The number of (a, b)-cores is the rational Catalan number 1 a + b Cat(a, b) := . a + b a

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Next goal: average size of an (a, b)-core.

Ehrhart theory

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Ehrhart theory

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 1 If V is an n-dimensional vector space, L is a lattice, P is a polytope with vertices in L and h is a polynomial of degree r, then for positive integers b X G(b) := h(x) x∈L∩bP is a polynomial of degree n + r in b. 2 Furthermore, we have X G( b) = ( 1)n h(x). − − x∈L∩−bP◦

Euler-Maclaurin theory

Theorem (Ehrhart, folklore)

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups 2 Furthermore, we have X G( b) = ( 1)n h(x). − − x∈L∩−bP◦

Euler-Maclaurin theory

Theorem (Ehrhart, folklore) 1 If V is an n-dimensional vector space, L is a lattice, P is a polytope with vertices in L and h is a polynomial of degree r, then for positive integers b X G(b) := h(x) x∈L∩bP is a polynomial of degree n + r in b.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Euler-Maclaurin theory

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups To calculate the average size of an (a, b)-core, recall that X X a ρ 2 ρ 2 size(λ) = µ . 2 − a − a λ (a,b)-core µ∈Qˇ∩Sa(b) Euler-Maclaurin theory implies that this is a polynomial of degree a + 1 in b. It is divisible by the polynomial of degree a 1 which counts the number of (a, b)-cores, so the average size of− an (a, b)-core is a polynomial of degree 2 in b. It is determined by its evaluation at 3 points.

The average size of an (a, b)-core

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups , recall that X X a ρ 2 ρ 2 size(λ) = µ . 2 − a − a λ (a,b)-core µ∈Qˇ∩Sa(b) Euler-Maclaurin theory implies that this is a polynomial of degree a + 1 in b. It is divisible by the polynomial of degree a 1 which counts the number of (a, b)-cores, so the average size of− an (a, b)-core is a polynomial of degree 2 in b. It is determined by its evaluation at 3 points.

The average size of an (a, b)-core

To calculate the average size of an (a, b)-core

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Euler-Maclaurin theory implies that this is a polynomial of degree a + 1 in b. It is divisible by the polynomial of degree a 1 which counts the number of (a, b)-cores, so the average size of− an (a, b)-core is a polynomial of degree 2 in b. It is determined by its evaluation at 3 points.

The average size of an (a, b)-core

To calculate the average size of an (a, b)-core, recall that X X a ρ 2 ρ 2 size(λ) = µ . 2 − a − a λ (a,b)-core µ∈Qˇ∩Sa(b)

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups It is divisible by the polynomial of degree a 1 which counts the number of (a, b)-cores, so the average size of− an (a, b)-core is a polynomial of degree 2 in b. It is determined by its evaluation at 3 points.

The average size of an (a, b)-core

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups , so the average size of an (a, b)-core is a polynomial of degree 2 in b. It is determined by its evaluation at 3 points.

The average size of an (a, b)-core

To calculate the average size of an (a, b)-core, recall that X X a ρ 2 ρ 2 size(λ) = µ . 2 − a − a λ (a,b)-core µ∈Qˇ∩Sa(b) Euler-Maclaurin theory implies that this is a polynomial of degree a + 1 in b. It is divisible by the polynomial of degree a 1 which counts the number of (a, b)-cores −

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups It is determined by its evaluation at 3 points.

The average size of an (a, b)-core

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The average size of an (a, b)-core

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Theorem (P. Johnson ’15, conjectured by Armstrong ’12) The average size of an (a, b)-core is

(a 1)(b 1)(a + b + 1) − − . 24

Conclusion:

No simple proof of this fact is known.

The average size of an (a, b)-core

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Theorem (P. Johnson ’15, conjectured by Armstrong ’12) The average size of an (a, b)-core is

(a 1)(b 1)(a + b + 1) − − . 24

No simple proof of this fact is known.

The average size of an (a, b)-core

Conclusion:

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups No simple proof of this fact is known.

The average size of an (a, b)-core

Conclusion: Theorem (P. Johnson ’15, conjectured by Armstrong ’12) The average size of an (a, b)-core is

(a 1)(b 1)(a + b + 1) − − . 24

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups No simple proof of this fact is known.

The average size of an (a, b)-core

Conclusion: Theorem (P. Johnson ’15, conjectured by Armstrong ’12) The average size of an (a, b)-core is

(a 1)(b 1)(a + b + 1) − − . 24

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The average size of an (a, b)-core

Conclusion: Theorem (P. Johnson ’15, conjectured by Armstrong ’12) The average size of an (a, b)-core is

(a 1)(b 1)(a + b + 1) − − . 24 No simple proof of this fact is known.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Theorem (M. Thiel and N. Williams ’15) The variance of the size of an (a, b)-core is

ab(a 1)(b 1)(a + b)(a + b + 1) − − . 1440

Similarly:

D. Zeilberger has computed some higher moments and conjectured a limiting distribution for the size of an (a, a + k)-core for ﬁxed k and a . → ∞

The average size of an (a, b)-core

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Theorem (M. Thiel and N. Williams ’15) The variance of the size of an (a, b)-core is

ab(a 1)(b 1)(a + b)(a + b + 1) − − . 1440

D. Zeilberger has computed some higher moments and conjectured a limiting distribution for the size of an (a, a + k)-core for ﬁxed k and a . → ∞

The average size of an (a, b)-core

Similarly:

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups D. Zeilberger has computed some higher moments and conjectured a limiting distribution for the size of an (a, a + k)-core for ﬁxed k and a . → ∞

The average size of an (a, b)-core

Similarly: Theorem (M. Thiel and N. Williams ’15) The variance of the size of an (a, b)-core is

ab(a 1)(b 1)(a + b)(a + b + 1) − − . 1440

The average size of an (a, b)-core

Similarly: Theorem (M. Thiel and N. Williams ’15) The variance of the size of an (a, b)-core is

ab(a 1)(b 1)(a + b)(a + b + 1) − − . 1440

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups and conjectured a limiting distribution for the size of an (a, a + k)-core for ﬁxed k and a . → ∞

The average size of an (a, b)-core

Similarly: Theorem (M. Thiel and N. Williams ’15) The variance of the size of an (a, b)-core is

ab(a 1)(b 1)(a + b)(a + b + 1) − − . 1440 D. Zeilberger has computed some higher moments

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The average size of an (a, b)-core

Similarly: Theorem (M. Thiel and N. Williams ’15) The variance of the size of an (a, b)-core is

ab(a 1)(b 1)(a + b)(a + b + 1) − − . 1440 D. Zeilberger has computed some higher moments and conjectured a limiting distribution for the size of an (a, a + k)-core for ﬁxed k and a . → ∞

Marko Thiel Simultaneous core partitions for affine Weyl groups Apart from the affine symmetric group, there are other affine reflection groups. Example: The affine Weyl group Wf of type Ce2.

Other aﬃne reﬂection groups

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Example: The aﬃne Weyl group Wf of type Ce2.

Other aﬃne reﬂection groups

Apart from the affine symmetric group, there are other affine reflection groups.

Marko Thiel Simultaneous core partitions for affine Weyl groups Other affine reflection groups

Apart from the affine symmetric group, there are other affine reflection groups. Example: The affine Weyl group Wf of type Ce2.

Marko Thiel Simultaneous core partitions for affine Weyl groups Other affine reflection groups

There is a lattice of cores (coroot lattice) Qˇ. Example:

Marko Thiel Simultaneous core partitions for affine Weyl groups Other affine reflection groups

For b relatively prime to the Coxeter number h of the aﬃne Weyl group Wf, there is a simplex (b). Example: (5) SWf SWf

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Then we can interpret them as special cases of statements that hold for any simply-laced aﬃne Weyl group Wf. Simply-laced: All angles between walls of the fundamental alcove A◦ are 60 or 90 degrees.

Other aﬃne reﬂection groups

We can understand statements about a-cores in terms of the ˇ lattice of cores Q of the aﬃne symmetric group Sea.

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Simply-laced: All angles between walls of the fundamental alcove A◦ are 60 or 90 degrees.

Other aﬃne reﬂection groups

We can understand statements about a-cores in terms of the ˇ lattice of cores Q of the aﬃne symmetric group Sea. Then we can interpret them as special cases of statements that hold for any simply-laced aﬃne Weyl group Wf.

Marko Thiel Simultaneous core partitions for affine Weyl groups Other affine reflection groups

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2 (a2 1)(b2 1)

max µ = − − . µ∈Qˇ∩Sa(b) 2 − a − a 24 The corresponding statement for any simply-laced aﬃne Weyl group Wf: h ρ 2 ρ 2 n(h + 1)(b2 1) max µ = − . µ∈Qˇ∩S (b) 2 − h − h 24 Wf

Here n is the dimension of Wf (a 1 for Sea) − and h is the Coxeter number (a for Sea).

The maximal size of an (a, b)-core

In terms of cores: (a2 1)(b2 1) The unique largest (a, b)-core has size − − . 24

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The corresponding statement for any simply-laced aﬃne Weyl group Wf: h ρ 2 ρ 2 n(h + 1)(b2 1) max µ = − . µ∈Qˇ∩S (b) 2 − h − h 24 Wf

Here n is the dimension of Wf (a 1 for Sea) − and h is the Coxeter number (a for Sea).

The maximal size of an (a, b)-core

In terms of cores: (a2 1)(b2 1) The unique largest (a, b)-core has size − − . 24

In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2 (a2 1)(b2 1)

max µ = − − . µ∈Qˇ∩Sa(b) 2 − a − a 24

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Here n is the dimension of Wf (a 1 for Sea) − and h is the Coxeter number (a for Sea).

The maximal size of an (a, b)-core

In terms of cores: (a2 1)(b2 1) The unique largest (a, b)-core has size − − . 24

In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2 (a2 1)(b2 1)

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups and h is the Coxeter number (a for Sea).

The maximal size of an (a, b)-core

In terms of cores: (a2 1)(b2 1) The unique largest (a, b)-core has size − − . 24

In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2 (a2 1)(b2 1)

Here n is the dimension of Wf (a 1 for Sea) −

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The maximal size of an (a, b)-core

In terms of cores: (a2 1)(b2 1) The unique largest (a, b)-core has size − − . 24

In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2 (a2 1)(b2 1)

Here n is the dimension of Wf (a 1 for Sea) − and h is the Coxeter number (a for Sea).

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2 (a 1)(b 1)(a + b + 1)

E µ = − − . µ∈Qˇ∩Sa(b) 2 − a − a 24 The corresponding statement for any simply-laced aﬃne Weyl group Wf: h ρ 2 ρ 2 n(b 1)(h + b + 1) E µ = − . µ∈Qˇ∩S (b) 2 − h − h 24 Wf

The average size of an (a, b)-core

In terms of cores: (a 1)(b 1)(a + b + 1) E (size(λ)) = − − . λ (a,b)-core 24

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The corresponding statement for any simply-laced aﬃne Weyl group Wf: h ρ 2 ρ 2 n(b 1)(h + b + 1) E µ = − . µ∈Qˇ∩S (b) 2 − h − h 24 Wf

The average size of an (a, b)-core

In terms of cores: (a 1)(b 1)(a + b + 1) E (size(λ)) = − − . λ (a,b)-core 24

In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2 (a 1)(b 1)(a + b + 1)

E µ = − − . µ∈Qˇ∩Sa(b) 2 − a − a 24

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The average size of an (a, b)-core

In terms of cores: (a 1)(b 1)(a + b + 1) E (size(λ)) = − − . λ (a,b)-core 24

In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2 (a 1)(b 1)(a + b + 1)

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2

V µ = RHS as above. µ∈Qˇ∩Sa(b) 2 − a − a The corresponding statement for any simply-laced aﬃne Weyl group Wf: h ρ 2 ρ 2 nhb(b 1)(h + b)(h + b + 1) V µ = − . µ∈Qˇ∩S (b) 2 − h − h 1440 Wf

The variance of the size of an (a, b)-core

In terms of cores: ab(a 1)(b 1)(a + b)(a + b + 1) V (size(λ)) = − − . λ (a,b)-core 1440

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The corresponding statement for any simply-laced aﬃne Weyl group Wf: h ρ 2 ρ 2 nhb(b 1)(h + b)(h + b + 1) V µ = − . µ∈Qˇ∩S (b) 2 − h − h 1440 Wf

The variance of the size of an (a, b)-core

In terms of cores: ab(a 1)(b 1)(a + b)(a + b + 1) V (size(λ)) = − − . λ (a,b)-core 1440

In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2

V µ = RHS as above. µ∈Qˇ∩Sa(b) 2 − a − a

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups The variance of the size of an (a, b)-core

In terms of cores: ab(a 1)(b 1)(a + b)(a + b + 1) V (size(λ)) = − − . λ (a,b)-core 1440

In terms of the lattice of cores for the aﬃne symmetric group Sea: a ρ 2 ρ 2

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups Thanks for your attention!

Marko Thiel Simultaneous core partitions for aﬃne Weyl groups