Combinatorics of the Cell Decomposition of Affine

Home , Affine symmetric group

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers

Michael Lugo

Virginia Tech Under the Advising of Mark Shimozono

28 April 2015 Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers

Contents

1 Aﬃne Springer Fiber

2 Aﬃne Symmetric Group

3 Hikita’s Representation

4 Gorsky, Mazin, and Vazirani’s Parking Functions

5 Combinatorial Connection Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Overview

1 Aﬃne Springer Fiber

2 Aﬃne Symmetric Group

3 Hikita’s Representation

4 Gorsky, Mazin, and Vazirani’s Parking Functions

5 Combinatorial Connection G := GLn(K) (alternatively SLn(K))

P := GLn(O) (alternatively SLn(O)) B := {g ∈ P | g evalulated at = 0 is uppertriangular} (Iwahori subgroup)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Notation

K := C(()) and O = C[[]] Fix n ≥ 1 V := K n be a K-vector space B := {g ∈ P | g evalulated at = 0 is uppertriangular} (Iwahori subgroup)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Notation

K := C(()) and O = C[[]] Fix n ≥ 1 V := K n be a K-vector space

G := GLn(K) (alternatively SLn(K))

P := GLn(O) (alternatively SLn(O)) Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Notation

K := C(()) and O = C[[]] Fix n ≥ 1 V := K n be a K-vector space

G := GLn(K) (alternatively SLn(K))

P := GLn(O) (alternatively SLn(O)) B := {g ∈ P | g evalulated at = 0 is uppertriangular} (Iwahori subgroup) (..., a−1, b−1, c−1, a0, b0, c0, a1, b1, c1,... )

acts by shifting n spots n n If {ei }i=1 is a C basis of C , then

k ei+kn = ei

is a C basis of V

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Interpretation

Consider elements of V as inﬁnite C tuples

X i X i X i ai bi ci

i≥N1 i≥N2 i≥N3 acts by shifting n spots n n If {ei }i=1 is a C basis of C , then

k ei+kn = ei

is a C basis of V

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Interpretation

Consider elements of V as inﬁnite C tuples

X i X i X i ai bi ci

i≥N1 i≥N2 i≥N3

(..., a−1, b−1, c−1, a0, b0, c0, a1, b1, c1,... ) n n If {ei }i=1 is a C basis of C , then

k ei+kn = ei

is a C basis of V

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Interpretation

Consider elements of V as inﬁnite C tuples

X i X i X i ai bi ci

i≥N1 i≥N2 i≥N3

(..., a−1, b−1, c−1, a0, b0, c0, a1, b1, c1,... )

acts by shifting n spots Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Interpretation

Consider elements of V as inﬁnite C tuples

X i X i X i ai bi ci

i≥N1 i≥N2 i≥N3

(..., a−1, b−1, c−1, a0, b0, c0, a1, b1, c1,... )

acts by shifting n spots n n If {ei }i=1 is a C basis of C , then

k ei+kn = ei

is a C basis of V Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Aﬃne Grassmannian

Deﬁnition ([1, 4])

The affine Grassmannian Gn for the group GLn is the moduli space of O-submodules M of V such that (a) M is O-invariant (b) M has rank n as a O-module (c) There exists N such that −N On ⊃ M ⊃ N On −N n N n (d) dimC O /M = dimC M/ O ∼ Note Gn = G/P. Combinatorics of the Cell Decomposition of Affine Springer Fibers Affine Springer Fiber

Aﬃne Flag

Deﬁnition

The aﬃne ﬂag ind-variety Fn is the space of collections

M0 ⊃ M1 ⊃ · · · ⊃ Mn = M0 such that

1 For all i, Mi satisﬁes (a), (b), and (c) for Gn

2 M0 ∈ Gn

3 dimC Mi /Mi+1 = 1 ∼ Note Fn = G/B Let m > n, gcd(m, n) = 1 and m = nk + b m T = N and In + T ∈ G Definition Let T be a semiregular nil-elliptic endomorphism. The Affine Springer Fiber is the set of fixed points of T over Fn. Equivalently, u if u = In + T , then the Affine Springer Fiber is F = Fm/n.

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Aﬃne Springer Fiber

Consider the single shift of the C basis of V 0 1 0 ... 0 0 0 1 ... 0    ..  N = 0 0 0 . 0   ......  . . . . . 0 0 ... 0 m T = N and In + T ∈ G Definition Let T be a semiregular nil-elliptic endomorphism. The Affine Springer Fiber is the set of fixed points of T over Fn. Equivalently, u if u = In + T , then the Affine Springer Fiber is F = Fm/n.

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Aﬃne Springer Fiber

Consider the single shift of the C basis of V 0 1 0 ... 0 0 0 1 ... 0    ..  N = 0 0 0 . 0   ......  . . . . . 0 0 ... 0

Let m > n, gcd(m, n) = 1 and m = nk + b Definition Let T be a semiregular nil-elliptic endomorphism. The Affine Springer Fiber is the set of fixed points of T over Fn. Equivalently, u if u = In + T , then the Affine Springer Fiber is F = Fm/n.

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Aﬃne Springer Fiber

Consider the single shift of the C basis of V 0 1 0 ... 0 0 0 1 ... 0    ..  N = 0 0 0 . 0   ......  . . . . . 0 0 ... 0

Let m > n, gcd(m, n) = 1 and m = nk + b m T = N and In + T ∈ G Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Springer Fiber

Aﬃne Springer Fiber

Consider the single shift of the C basis of V 0 1 0 ... 0 0 0 1 ... 0    ..  N = 0 0 0 . 0   ......  . . . . . 0 0 ... 0

Let m > n, gcd(m, n) = 1 and m = nk + b m T = N and In + T ∈ G Definition Let T be a semiregular nil-elliptic endomorphism. The Affine Springer Fiber is the set of fixed points of T over Fn. Equivalently, u if u = In + T , then the Affine Springer Fiber is F = Fm/n. Combinatorics of the Cell Decomposition of Affine Springer Fibers Affine Symmetric Group

Overview

1 Aﬃne Springer Fiber

2 Aﬃne Symmetric Group

3 Hikita’s Representation

4 Gorsky, Mazin, and Vazirani’s Parking Functions

5 Combinatorial Connection Let Hi ⊂ E be the hyperplane perpendicular to αi {Hi } acts on E by reflection Define si to be the reflection across Hi

The complement of {Hi }, closed under reﬂections, form Weyl chambers

The fundamental chamber is on the positive side of all Hi Assigning 1 ∈ Sn to the fundamental chamber creates an isomorphism to Sn

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Symmetric Group

n Let E ⊂ R be perpendicular to (1, 1,..., 1) Let αi be the ith simple coroot (0, 0,..., 1, −1 ..., 0) with 1 in the ith position, 1 ≤ i < n The complement of {Hi }, closed under reﬂections, form Weyl chambers

The fundamental chamber is on the positive side of all Hi Assigning 1 ∈ Sn to the fundamental chamber creates an isomorphism to Sn

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Symmetric Group

n Let E ⊂ R be perpendicular to (1, 1,..., 1) Let αi be the ith simple coroot (0, 0,..., 1, −1 ..., 0) with 1 in the ith position, 1 ≤ i < n

Let Hi ⊂ E be the hyperplane perpendicular to αi {Hi } acts on E by reflection Define si to be the reflection across Hi Combinatorics of the Cell Decomposition of Affine Springer Fibers Affine Symmetric Group

Aﬃne Symmetric Group

n Let E ⊂ R be perpendicular to (1, 1,..., 1) Let αi be the ith simple coroot (0, 0,..., 1, −1 ..., 0) with 1 in the ith position, 1 ≤ i < n

Let Hi ⊂ E be the hyperplane perpendicular to αi {Hi } acts on E by reflection Define si to be the reflection across Hi

The complement of {Hi }, closed under reﬂections, form Weyl chambers

The fundamental chamber is on the positive side of all Hi Assigning 1 ∈ Sn to the fundamental chamber creates an isomorphism to Sn H1 H2

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Example n = 3

α2 α1 Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Example n = 3

H1 H2

α2 α1 α2 α1

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Example n = 3

H1 H2 H1 H2

α2 α1

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Example n = 3 H1 H2

α2 α1

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Example n = 3

1 H1 H2

α2 α1

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Example n = 3

s1 s2 H1 H2

α2 α1

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Example n = 3

s1 s2

s1s2 s2s1

s2s1s2 s0s1s2s1

s0s1s2 s0s2s1

s0s1 s0s2 s1s0s1s2 s2s0s2s1 s0 s1s0s1s2s1 s1s0s1 s2s0s2 s2s0s2s1s2

1 H0

s1s0s2s1 s1s0 s2s0 s2s0s1s2 1 s1s0s2 s2s0s1 s1 s2

s1s2 s2s1

s2s1s2

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Symmetric Group (n = 3) s0s1s2s1

s0s1s2 s0s2s1

s0s1 s0s2 s1s0s1s2 s2s0s2s1 s0 s1s0s1s2s1 s1s0s1 s2s0s2 s2s0s2s1s2

s1s0s2s1 s1s0 s2s0 s2s0s1s2 1 s1s0s2 s2s0s1 s1 s2

s1s2 s2s1

s2s1s2

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Symmetric Group (n = 3)

1 H0 s0s1s2s1

s0s1s2 s0s2s1

s0s1 s0s2 s1s0s1s2 s2s0s2s1 s0 s1s0s1s2s1 s1s0s1 s2s0s2 s2s0s2s1s2

s1s0s2s1 s1s0 s2s0 s2s0s1s2

s1s0s2 s2s0s1

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Symmetric Group (n = 3)

1 H0

s1 s2

s1s2 s2s1

s2s1s2 s0s1s2s1

s0s1s2 s0s2s1

s0s1 s0s2 s1s0s1s2 s2s0s2s1

s1s0s1s2s1 s1s0s1 s2s0s2 s2s0s2s1s2

1 H0

s1s0s2s1 s1s0 s2s0 s2s0s1s2

s1s0s2 s2s0s1

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Symmetric Group (n = 3)

s1 s2

s1s2 s2s1

s2s1s2 s0s1s2s1

s0s1s2 s0s2s1

s0s1 s0s2 s1s0s1s2 s2s0s2s1

s1s0s1s2s1 s1s0s1 s2s0s2 s2s0s2s1s2

1 H0

s1s0s2s1 s1s0 s2s0 s2s0s1s2

s1s0s2 s2s0s1

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Symmetric Group (n = 3)

s1 s2

s1s2 s2s1

s2s1s2 1 H0

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Symmetric Group (n = 3)

s0s1s2s1

s0s1s2 s0s2s1

s0s1 s0s2 s1s0s1s2 s2s0s2s1 s0 s1s0s1s2s1 s1s0s1 s2s0s2 s2s0s2s1s2

s1s0s2s1 s1s0 s2s0 s2s0s1s2 1 s1s0s2 s2s0s1 s1 s2

s1s2 s2s1

s2s1s2 Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Symmetric Group

Deﬁnition (Aﬃne Symmetric Group)

The aﬃne symmetric group Sen is the group of words in {s0,..., sn−1} with the following relations: 2 si = 1 si sj si = sj si sj if i − j 6≡ ±1 (mod n)

si sj = sj si if i − j ≡ ±1 (mod n) and n > 2 ∨ n Pn Q := {(x1,..., xn) ∈ Z | i=1 xi = 0} ∼ ∨ Sen = Q o Sn If ω ∈ Sen is aﬃne grassmannian, then it is the identity or all its reduced words end with s0

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Grassmannians

Consider the cosets of Sen/Sn

Deﬁnition

ω ∈ Sen is aﬃne grassmannian if it is the minimum length representative of ωSn in Sen/Sn. If ω ∈ Sen is aﬃne grassmannian, then it is the identity or all its reduced words end with s0

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Grassmannians

Consider the cosets of Sen/Sn

Deﬁnition

ω ∈ Sen is aﬃne grassmannian if it is the minimum length representative of ωSn in Sen/Sn.

∨ n Pn Q := {(x1,..., xn) ∈ Z | i=1 xi = 0} ∼ ∨ Sen = Q o Sn Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Aﬃne Grassmannians

Consider the cosets of Sen/Sn

Deﬁnition

ω ∈ Sen is aﬃne grassmannian if it is the minimum length representative of ωSn in Sen/Sn.

∨ n Pn Q := {(x1,..., xn) ∈ Z | i=1 xi = 0} ∼ ∨ Sen = Q o Sn If ω ∈ Sen is affine grassmannian, then it is the identity or all its reduced words end with s0 Combinatorics of the Cell Decomposition of Affine Springer Fibers Affine Symmetric Group Combinatorics of the Cell Decomposition of Affine Springer Fibers Affine Symmetric Group Combinatorics of the Cell Decomposition of Affine Springer Fibers Affine Symmetric Group Combinatorics of the Cell Decomposition of Affine Springer Fibers Affine Symmetric Group   ω(x + 1) if x ≡ i (mod n) (ωsi )(x) = ω(x − 1) if x ≡ i + 1 (mod n)  ω(x) otherwise

n X n(n + 1) ω(i) = 2 i=1 ω(x + n) = ω(x) + n Denoted by window notation [ω(1), ω(2), . . . , ω(n)]

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

The Permutation

Consider ω ∈ Sen as a permutation Z → Z

1 7→ idZ We need only deﬁne the si action n X n(n + 1) ω(i) = 2 i=1 ω(x + n) = ω(x) + n Denoted by window notation [ω(1), ω(2), . . . , ω(n)]

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

The Permutation

Consider ω ∈ Sen as a permutation Z → Z

1 7→ idZ We need only define the si action   ω(x + 1) if x ≡ i (mod n) (ωsi )(x) = ω(x − 1) if x ≡ i + 1 (mod n)  ω(x) otherwise Combinatorics of the Cell Decomposition of Affine Springer Fibers Affine Symmetric Group

The Permutation

Consider ω ∈ Sen as a permutation Z → Z

1 7→ idZ We need only deﬁne the si action   ω(x + 1) if x ≡ i (mod n) (ωsi )(x) = ω(x − 1) if x ≡ i + 1 (mod n)  ω(x) otherwise

n X n(n + 1) ω(i) = 2 i=1 ω(x + n) = ω(x) + n Denoted by window notation [ω(1), ω(2), . . . , ω(n)] G Fn = BwB/B

ω∈Sen G Gn = BwP/P

ω∈Sen/Sn

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Indexing Sets

Recall Fn = G/B and Gn = G/P G Gn = BwP/P

ω∈Sen/Sn

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Indexing Sets

Recall Fn = G/B and Gn = G/P G Fn = BwB/B

ω∈Sen Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Aﬃne Symmetric Group

Indexing Sets

Recall Fn = G/B and Gn = G/P G Fn = BwB/B

ω∈Sen G Gn = BwP/P

ω∈Sen/Sn Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Overview

1 Aﬃne Springer Fiber

2 Aﬃne Symmetric Group

3 Hikita’s Representation

4 Gorsky, Mazin, and Vazirani’s Parking Functions

5 Combinatorial Connection λ Fm/n ⊆ Fm/n ⊆

π−1 π

Cλ ⊆ Gm/n ⊆

(Gm/n := π(Fm/n))

Gn has paving by Iwahori orbits BωP/P where ω ∈ Sen/Sn

Gm/n has a paving by Gm/n ∩ (BωP/P)

The non-zero cells are indexed by partitions, called Cλ λ −1 Fm/n = π (Cλ)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Notation

Fn π

π : Fn → Gn the natural projection λ Fm/n ⊆

π−1

Cλ ⊆

Gn has paving by Iwahori orbits BωP/P where ω ∈ Sen/Sn

Gm/n has a paving by Gm/n ∩ (BωP/P)

The non-zero cells are indexed by partitions, called Cλ λ −1 Fm/n = π (Cλ)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Notation

Fm/n ⊆ Fn π π

Gm/n ⊆ Gn

π : Fn → Gn the natural projection (Gm/n := π(Fm/n)) λ Fm/n ⊆

π−1

Cλ ⊆

The non-zero cells are indexed by partitions, called Cλ λ −1 Fm/n = π (Cλ)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Notation

Fm/n ⊆ Fn π π

Gm/n ⊆ Gn

π : Fn → Gn the natural projection (Gm/n := π(Fm/n))

Gn has paving by Iwahori orbits BωP/P where ω ∈ Sen/Sn

Gm/n has a paving by Gm/n ∩ (BωP/P) λ Fm/n ⊆

π−1 ⊆

λ −1 Fm/n = π (Cλ)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Notation

Fm/n ⊆ Fn π π

Cλ ⊆ Gm/n ⊆ Gn

π : Fn → Gn the natural projection (Gm/n := π(Fm/n))

Gn has paving by Iwahori orbits BωP/P where ω ∈ Sen/Sn

Gm/n has a paving by Gm/n ∩ (BωP/P)

The non-zero cells are indexed by partitions, called Cλ Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Notation

λ Fm/n ⊆ Fm/n ⊆ Fn

π−1 π π

Cλ ⊆ Gm/n ⊆ Gn

π : Fn → Gn the natural projection (Gm/n := π(Fm/n))

Gn has paving by Iwahori orbits BωP/P where ω ∈ Sen/Sn

Gm/n has a paving by Gm/n ∩ (BωP/P)

The non-zero cells are indexed by partitions, called Cλ λ −1 Fm/n = π (Cλ) Pn−1 a = i=1 ai sa+l · λˇ(a1,..., an−1) =  λˇ(a1,..., al+1, al ,..., an−1) if l = 1, 2,..., n − 2,   λˇ(an−1 − 1, a1,..., an−2) if l = n − 1 and an−1 ≥ 1, λˇ(a ,..., a ) if l = n − 1 and a = 0,  1 n−1 n−1  λˇ(a2,..., an−1, a1 + 1) if l = 0.

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Hikita Representation

≥0 P = {(ai ) | ai ∈ Z , an = 0, a1,..., an−1 ≥ 0, ai+n = ai + 1} λˇ : P → Q∨ is a bijection Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Hikita Representation

≥0 P = {(ai ) | ai ∈ Z , an = 0, a1,..., an−1 ≥ 0, ai+n = ai + 1} λˇ : P → Q∨ is a bijection Pn−1 a = i=1 ai sa+l · λˇ(a1,..., an−1) =  λˇ(a1,..., al+1, al ,..., an−1) if l = 1, 2,..., n − 2,   λˇ(an−1 − 1, a1,..., an−2) if l = n − 1 and an−1 ≥ 1, λˇ(a ,..., a ) if l = n − 1 and a = 0,  1 n−1 n−1  λˇ(a2,..., an−1, a1 + 1) if l = 0. Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Example

(3,0) (0,3)

(2,1) (2,0) (1,2) (0,2)

(1,1) (1,0) (0,1)

(0,0) δ = (k(n − 1) + b − 1, k(n − 2) + b − 1,..., k + b − 1, b − 1) δ − δ0 is the partition below the diagonal (0, n) to (m, 0)

Proposition ([2]) There is a bijection n o (a ) ∈ P | C 6= ∅ → {λ | λ is a partition, λ ⊆ δ − δ0} i λˇ(ai )

(ai ) 7→ λ(ai ) := δ − (ab, a2b,..., anb)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Relation to Partitions

Recall m = nk + b Proposition ([2]) There is a bijection n o (a ) ∈ P | C 6= ∅ → {λ | λ is a partition, λ ⊆ δ − δ0} i λˇ(ai )

(ai ) 7→ λ(ai ) := δ − (ab, a2b,..., anb)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Relation to Partitions

Recall m = nk + b δ = (k(n − 1) + b − 1, k(n − 2) + b − 1,..., k + b − 1, b − 1) δ − δ0 is the partition below the diagonal (0, n) to (m, 0) Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Hikita’s Representation

Relation to Partitions

Recall m = nk + b δ = (k(n − 1) + b − 1, k(n − 2) + b − 1,..., k + b − 1, b − 1) δ − δ0 is the partition below the diagonal (0, n) to (m, 0)

Proposition ([2]) There is a bijection n o (a ) ∈ P | C 6= ∅ → {λ | λ is a partition, λ ⊆ δ − δ0} i λˇ(ai )

(ai ) 7→ λ(ai ) := δ − (ab, a2b,..., anb) Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Overview

1 Aﬃne Springer Fiber

2 Aﬃne Symmetric Group

3 Hikita’s Representation

4 Gorsky, Mazin, and Vazirani’s Parking Functions

5 Combinatorial Connection −→ 2141 L M

f :[n] → [n] such that f (i) is the ith car’s parking preference A car will go to its preference, then take the next open spot f is a parking function if all can park without circling back Denote a parking function by f (1) f (2) f (3) ... f (n) L M Consider f (1) = 2, f (2) = 1, f (3) = 4, f (4) = 1

1 2 3 4

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Functions

Denote [n] = {1, 2,..., n} Let there be n parking spots on a one-way street −→ 2141 L M

Denote a parking function by f (1) f (2) f (3) ... f (n) L M Consider f (1) = 2, f (2) = 1, f (3) = 4, f (4) = 1

1 2 3 4

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Functions

Denote [n] = {1, 2,..., n} Let there be n parking spots on a one-way street f :[n] → [n] such that f (i) is the ith car’s parking preference A car will go to its preference, then take the next open spot f is a parking function if all can park without circling back −→ 2141 L M

Consider f (1) = 2, f (2) = 1, f (3) = 4, f (4) = 1

1 2 3 4

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Functions

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Functions

1 2 3 4 −→ 2141 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Functions

1 2 3 4 −→ 2141 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Functions

2 1

1 2 3 4 −→ 2141 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Functions

2 1 3

1 2 3 4 −→ 2141 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Functions

2 1 4 3

1 2 3 4 Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Functions

2 1 4 3

1 2 3 4 Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Parking Function Properties

f ([n]), when sorted as a1, a2,..., an, obeys ai ≤ i Any permutation of a parking function is a parking function There are (n + 1)n−1 parking functions on domain [n] 2040 6∈ PF 5/4 but 2040 ∈ PF 7/4 L M L M

n−1 |PF m/n| = m Consider the parking function 2040 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

PF m/n

Let n < m and gcd(m, n) = 1

PF m/n - Parking functions whose Young diagram ﬁts below the diagonal of an n × m box 2040 6∈ PF 5/4 but 2040 ∈ PF 7/4 L M L M

Consider the parking function 2040 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

PF m/n

Let n < m and gcd(m, n) = 1

PF m/n - Parking functions whose Young diagram ﬁts below the diagonal of an n × m box n−1 |PF m/n| = m 2040 6∈ PF 5/4 but 2040 ∈ PF 7/4 L M L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

PF m/n

Let n < m and gcd(m, n) = 1

PF m/n - Parking functions whose Young diagram ﬁts below the diagonal of an n × m box n−1 |PF m/n| = m Consider the parking function 2040 L M 2040 6∈ PF 5/4 but 2040 ∈ PF 7/4 L M L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

PF m/n

Let n < m and gcd(m, n) = 1

PF m/n - Parking functions whose Young diagram ﬁts below the diagonal of an n × m box n−1 |PF m/n| = m Consider the parking function 2040 L M but 2040 ∈ PF 7/4 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

PF m/n

Let n < m and gcd(m, n) = 1

PF m/n - Parking functions whose Young diagram ﬁts below the diagonal of an n × m box n−1 |PF m/n| = m Consider the parking function 2040 L M

2040 6∈ PF 5/4 L M but 2040 ∈ PF 7/4 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

PF m/n

Let n < m and gcd(m, n) = 1

PF m/n - Parking functions whose Young diagram ﬁts below the diagonal of an n × m box n−1 |PF m/n| = m Consider the parking function 2040 L M

2040 6∈ PF 5/4 L M Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

PF m/n

Let n < m and gcd(m, n) = 1

PF m/n - Parking functions whose Young diagram ﬁts below the diagonal of an n × m box n−1 |PF m/n| = m Consider the parking function 2040 L M

2040 6∈ PF 5/4 but 2040 ∈ PF 7/4 L M L M m GMV created a map SP : Sen → PF m/n

ω 7→ SPω

Proven to be a bijection for m = kn ± 1 SP is conjectured to be a bijection for all m

SPω(i) = #{j > i | 0 < ω(i) − ω(j) < m}

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Mapping

m Sen ⊂ Sen is the set of m-restricted permutations m Sen = {ω | i < j ⇒ ω(j) − ω(i) 6= m} Proven to be a bijection for m = kn ± 1 SP is conjectured to be a bijection for all m

SPω(i) = #{j > i | 0 < ω(i) − ω(j) < m}

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Mapping

m Sen ⊂ Sen is the set of m-restricted permutations m Sen = {ω | i < j ⇒ ω(j) − ω(i) 6= m} m GMV created a map SP : Sen → PF m/n

ω 7→ SPω SPω(i) = #{j > i | 0 < ω(i) − ω(j) < m}

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Mapping

m Sen ⊂ Sen is the set of m-restricted permutations m Sen = {ω | i < j ⇒ ω(j) − ω(i) 6= m} m GMV created a map SP : Sen → PF m/n

ω 7→ SPω

Proven to be a bijection for m = kn ± 1 SP is conjectured to be a bijection for all m Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Mapping

m Sen ⊂ Sen is the set of m-restricted permutations m Sen = {ω | i < j ⇒ ω(j) − ω(i) 6= m} m GMV created a map SP : Sen → PF m/n

ω 7→ SPω

Proven to be a bijection for m = kn ± 1 SP is conjectured to be a bijection for all m

SPω(i) = #{j > i | 0 < ω(i) − ω(j) < m} 3 0 1 1

SPω(1) =

SPω(2) =

SPω(3) =

SPω(4) =

SPω = 3011 L M

··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 3 5 8 2 7 9 ···

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] 3 0 1 1

SPω(1) =

SPω(2) =

SPω(3) =

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 3 5 8 2 7 9 ··· 0 1 1

SPω(2) =

SPω(3) =

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 3 5 8 2 7 9 ···

SPω(1) = 0 1 1

SPω(2) =

SPω(3) =

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 3 5 8 2 7 9 ···

SPω(1) = 0 1 1

SPω(2) =

SPω(3) =

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -235 8279 ···

SPω(1) = 0 1 1

SPω(2) =

SPω(3) =

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -235 8279 ···

SPω(1) = 3 1 1

SPω(3) =

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -235 8 2 7 9 ···

SPω(1) = 3

SPω(2) = 1 1

SPω(3) =

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -235 8279 ···

SPω(1) = 3

SPω(2) = 0 1

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -235 8 2 7 9 ···

SPω(1) = 3

SPω(2) = 0

SPω(3) = 1

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 35 8279 ···

SPω(1) = 3

SPω(2) = 0

SPω(3) = 1

SPω(4) =

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 35 8279 ···

SPω(1) = 3

SPω(2) = 0

SPω(3) = 1 1

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 35 8 2 7 9 ···

SPω(1) = 3

SPω(2) = 0

SPω(3) = 1

SPω(4) = 1

SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 3 5 8279 ···

SPω(1) = 3

SPω(2) = 0

SPω(3) = 1

SPω(4) = SPω = 3011 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 3 5 8279 ···

SPω(1) = 3

SPω(2) = 0

SPω(3) = 1

SPω(4) = 1 Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Example

SPω(i) = |{j > i | ω(i) − m < ω(j) < ω(i)}|

n = 4, m = 7, ω = [4, −2, 3, 5] ··· -3 -2 -1 0 1 2 3 4 5 6 7 8 ··· ··· 0 -6 -1 1 4 -2 3 5 8 2 7 9 ···

SPω(1) = 3

SPω(2) = 0

SPω(3) = 1

SPω(4) = 1

SPω = 3011 L M Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Relation to the Aﬃne Springer Fiber

Theorem ([3]) Consider the nil-elliptic operator T , where m is coprime to n. Then the corresponding affine Springer Fiber Fm/n ⊂ Fn admits an affine paving by mn−1 affine cells. Combinatorics of the Cell Decomposition of Affine Springer Fibers Gorsky, Mazin, and Vazirani’s Parking Functions

Theorem (GMV [1])

There is a natural bijection between the affine cells in Fm/n and m the affine permutations in Sen. The dimension of the cell Σω labeled by the affine permutation ω is equal to

n X SPω(i). i=1 Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Overview

1 Aﬃne Springer Fiber

2 Aﬃne Symmetric Group

3 Hikita’s Representation

4 Gorsky, Mazin, and Vazirani’s Parking Functions

5 Combinatorial Connection m Sen bijects with PF m/n

Non-zero Sen/Sn bijects with P π Fm/n −→Gm/n

How to map from PF m/n → P?

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

What do we know?

m Fm/n is paved by Sen

Gm/n is paved by a subset of Sen/Sn π Fm/n −→Gm/n

How to map from PF m/n → P?

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

What do we know?

m Fm/n is paved by Sen

Gm/n is paved by a subset of Sen/Sn m Sen bijects with PF m/n

Non-zero Sen/Sn bijects with P How to map from PF m/n → P?

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

What do we know?

m Fm/n is paved by Sen

Gm/n is paved by a subset of Sen/Sn m Sen bijects with PF m/n

Non-zero Sen/Sn bijects with P π Fm/n −→Gm/n Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

What do we know?

m Fm/n is paved by Sen

Gm/n is paved by a subset of Sen/Sn m Sen bijects with PF m/n

Non-zero Sen/Sn bijects with P π Fm/n −→Gm/n

How to map from PF m/n → P? (ai ) = δ − λ (0,0) (0,1) (1,0) (1,1) (2,1) PF(n+1)/n (2,1,0) (2,0,0) (1,1,0) (1,0,0) (0,0,0) (2,0,1) (0,2,0) (1,0,1) (0,1,0) (0,2,1) (0,0,2) (0,1,1) (0,0,1) (1,2,0) (0,1,2) (1,0,2)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

m = n + 1 and λ ⊆ δ

λ ∅ (2,1,0) (2,0,0) (1,1,0) (1,0,0) (0,0,0) PF(n+1)/n (2,1,0) (2,0,0) (1,1,0) (1,0,0) (0,0,0) (2,0,1) (0,2,0) (1,0,1) (0,1,0) (0,2,1) (0,0,2) (0,1,1) (0,0,1) (1,2,0) (0,1,2) (1,0,2)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

m = n + 1 and λ ⊆ δ

λ ∅ (2,1,0) (2,0,0) (1,1,0) (1,0,0) (0,0,0) (ai ) = δ − λ (0,0) (0,1) (1,0) (1,1) (2,1) Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

m = n + 1 and λ ⊆ δ

λ ∅ (2,1,0) (2,0,0) (1,1,0) (1,0,0) (0,0,0) (ai ) = δ − λ (0,0) (0,1) (1,0) (1,1) (2,1) PF(n+1)/n (2,1,0) (2,0,0) (1,1,0) (1,0,0) (0,0,0) (2,0,1) (0,2,0) (1,0,1) (0,1,0) (0,2,1) (0,0,2) (0,1,1) (0,0,1) (1,2,0) (0,1,2) (1,0,2) (0,1)

(1,1) (0,0) (1,0)

(2,1)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

Convert (ai ) to Sen (0,0) (0,1) (1,0) (1,1) (2,1) 1 s0 s2s0 s1s0 s2s1s2s0

(2,1)

(1,1) (1,0) (0,1)

(0,0) (2,1)

(1,1) (1,0) (0,1)

(0,0)

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

Convert (ai ) to Sen (0,0) (0,1) (1,0) (1,1) (2,1) 1 s0 s2s0 s1s0 s2s1s2s0

(0,1)

(1,1) (0,0) (1,0)

(2,1) k k Hi 7→ Hi,i+1 for 1 ≤ i < n k k H0 7→ H1,n k −k k k−1 Hi,j = Hj,i = Hi+tn,j+tn = Hi,j−n m 0 Dn is the Sommers region bounded by {Hi,i+m | 1 ≤ i ≤ n}

Lemma (GMV) m The set of alcoves {ω(A0) | ω ∈ Sen} coincides with the set of m alcoves that ﬁt inside the region Dn .

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Useful Facts

k GMV extended the Hi notation

k Hi,j = {x¯ ∈ E | xi − xj = k} k −k k k−1 Hi,j = Hj,i = Hi+tn,j+tn = Hi,j−n m 0 Dn is the Sommers region bounded by {Hi,i+m | 1 ≤ i ≤ n}

Lemma (GMV) m The set of alcoves {ω(A0) | ω ∈ Sen} coincides with the set of m alcoves that ﬁt inside the region Dn .

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Useful Facts

k GMV extended the Hi notation

k Hi,j = {x¯ ∈ E | xi − xj = k}

k k Hi 7→ Hi,i+1 for 1 ≤ i < n k k H0 7→ H1,n m 0 Dn is the Sommers region bounded by {Hi,i+m | 1 ≤ i ≤ n}

Lemma (GMV) m The set of alcoves {ω(A0) | ω ∈ Sen} coincides with the set of m alcoves that ﬁt inside the region Dn .

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Useful Facts

k GMV extended the Hi notation

k Hi,j = {x¯ ∈ E | xi − xj = k}

k k Hi 7→ Hi,i+1 for 1 ≤ i < n k k H0 7→ H1,n k −k k k−1 Hi,j = Hj,i = Hi+tn,j+tn = Hi,j−n Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Useful Facts

k GMV extended the Hi notation

k Hi,j = {x¯ ∈ E | xi − xj = k}

k k Hi 7→ Hi,i+1 for 1 ≤ i < n k k H0 7→ H1,n k −k k k−1 Hi,j = Hj,i = Hi+tn,j+tn = Hi,j−n m 0 Dn is the Sommers region bounded by {Hi,i+m | 1 ≤ i ≤ n}

Lemma (GMV) m The set of alcoves {ω(A0) | ω ∈ Sen} coincides with the set of m alcoves that ﬁt inside the region Dn . s1s0s1s2 s2s0s2s1 [150]120 [204]s1010s1 [042]s0200s2 [4-13]201 L Ms[105]1102s0s1L M[024]001s0 L Ms[-143]2021s0s2L M L M L M L M [015] [123] [-134] s0021s0 0001 s0112s0 L M[213]100s1 L M[132]010s2 L M L M L M

[231]s1101s2 [312]s2002s1 L M L M s[321]2210s1s2 L M s2[-226]s0121s2s0 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

4 Label D3 s1s0s1s2 s2s0s2s1 [150]120 [204]s1010s1 [042]s0200s2 [4-13]201 L Ms[105]1102s0s1L M[024]001s0 L Ms[-143]2021s0s2L M L M L M L M [015] [123] [-134] s0021s0 0001 s0112s0 L M[213]100s1 L M[132]010s2 L M L M L M

[231]s1101s2 [312]s2002s1 L M L M s[321]2210s1s2 L M s2[-226]s0121s2s0 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

4 Label D3 [150]120 [204]101 [042]020 [4-13]201 L M[105]102 L M[024]001 L M[-143]021 L M L M L M L M

[015]002 [123]000 [-134]011 L M[213]100 L M[132]010 L M L M L M [231]110 [312]200 L M[321]210 L M L M [-226]012 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

4 Label D3

s1s0s1s2 s2s0s2s1 s0s1 s0s2 s1s0s1 s0 s2s0s2

s1s0 1 s2s0 s1 s2

s1s2 s2s1 s2s1s2

s2s1s2s0 [150]120 [204]101 [042]020 [4-13]201 L M[105]102 L M[024]001 L M[-143]021 L M L M L M L M

[015]002 [123]000 [-134]011 L M[213]100 L M[132]010 L M L M L M [231]110 [312]200 L M[321]210 L M L M [-226]012 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

Convert to windows

s1s0s1s2 s2s0s2s1 s0s1 s0s2 s1s0s1 s0 s2s0s2

s1s0 1 s2s0 s1 s2

s1s2 s2s1 s2s1s2

s2s1s2s0 s1s0s1s2 s2s0s2s1 120 s1010s1 s0200s2 201 L Ms1102s0s1L M 001s0 L Ms2021s0s2L M L M L M L M

s0021s0 0001 s0112s0 L M 100s1 L M 010s2 L M L M L M

s1101s2 s2002s1 L M L M s2210s1s2 L M s2s0121s2s0 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

Convert to windows

[150] [204] [042] [4-13] [105] [024] [-143]

[015] [123] [-134] [213] [132]

[231] [312] [321] [-226] s1s0s1s2 s2s0s2s1 120 s1010s1 s0200s2 201 L Ms1102s0s1L M 001s0 L Ms2021s0s2L M L M L M L M

s0021s0 0001 s0112s0 L M 100s1 L M 010s2 L M L M L M

s1101s2 s2002s1 L M L M s2210s1s2 L M s2s0121s2s0 L M

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

Apply SP

[150] [204] [042] [4-13] [105] [024] [-143]

[015] [123] [-134] [213] [132]

[231] [312] [321] [-226] s1s0s1s2 s2s0s2s1 [150] [204]s0s1 [042]s0s2 [4-13] s[105]1s0s1 [024]s0 s[-143]2s0s2

[015] [123] [-134] s1s0 1 s2s0 [213]s1 [132]s2

[231]s1s2 [312]s2s1 s[321]2s1s2

s2[-226]s1s2s0

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Extended Example

Apply SP

120 101 020 201 L M 102 L M 001 L M 021 L M L M L M L M

002 000 011 L M 100 L M 010 L M L M L M 110 200 L M 210 L M L M 012 L M Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

The Projection

120 101 020 201 L M 102 L M 001 L M 021 L M (0,1) L M L M L M

002 000 011 (1,1) (0,0) (1,0) L M 100 L M 010 L M L M L M 110 200 L M 210 L M L M 012 L M (2,1) Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

The Un-natural Mapping

What is the combinatorial projection from PF 4/3 → P?

P (0,0) (0,1) (1,1) (1,0) (2,1) PF m/n 000 001 002 011 012 L100M L101M L102M L021M L M L010M L020M L120M L201M L110M L M L M L M L200M L210M L M A m PF Sen m/n

PS ω 7→ ω−1

m PF m/n mS D SP en n

Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

The Conjectured Natural Mapping

m GMV also created A : Sen → PF m/n a bijection Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

The Conjectured Natural Mapping

m GMV also created A : Sen → PF m/n a bijection

A m PF Sen m/n

PS ω 7→ ω−1

m PF m/n mS D SP en n Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

The Conjectured Natural Mapping

m-Restricted m-Stable A map (PF) Word of Restricted [1, 2, 3] [1, 2, 3] [0, 1, 2] [] [2, 1, 3] [2, 1, 3] [1, 0, 2] [1] [1, 3, 2] [1, 3, 2] [0, 2, 1] [2] [2, 3, 1] [3, 1, 2] [1, 2, 0] [1, 2] [3, 1, 2] [2, 3, 1] [2, 0, 1] [2, 1] [3, 2, 1] [3, 2, 1] [2, 1, 0] [1, 2, 1] [0, 2, 4] [0, 2, 4] [0, 2, 0] [0] [2, 0, 4] [0, 1, 5] [2, 0, 0] [0, 1] [0, 4, 2] [−1, 3, 4] [0, 0, 2] [0, 2] [0, 1, 5] [2, 0, 4] [0, 1, 1] [1, 0] [1, 0, 5] [1, 0, 5] [1, 0, 1] [1, 0, 1] [1, 5, 0] [1, −1, 6] [1, 1, 0] [1, 0, 1, 2] [−1, 3, 4] [0, 4, 2] [0, 0, 1] [2, 0] [−1, 4, 3] [−1, 4, 3] [0, 1, 0] [2, 0, 2] [4, −1, 3] [−2, 5, 3] [1, 0, 0] [2, 0, 2, 1] [−2, 2, 6] [4, 2, 0] [0, 0, 0] [2, 1, 2, 0] Combinatorics of the Cell Decomposition of Aﬃne Springer Fibers Combinatorial Connection

Eugene Gorsky, Makhail Mazin, and Monica Vazirani. Affine permutations and rational slope parking functions. arXiv, March 2014. Tatsuyuki Hikita. Affine srpinger fibers of type a and combinatorics of diagonal coinvariants. arXiv, 2012. G. Lusztig and J. M. Smelt. Fixed point varieties on the space of lattices. London Mathematical Society, 23:213–218, 1991. Peter Magyar. Affine schubert varieties and circular complexes. arXiv, 1999.