Combinatorics of the Cell Decomposition of Affine Springer Fibers

Transcription

1 Combinatorics of the Cell Decomposition of Affine Springer Fibers Michael Lugo Virginia Tech Under the Advising of Mark Shimozono 28 April 2015

2 Contents 1 Affine Springer Fiber 2 Affine Symmetric Group 3 Hikita s Representation 4 Gorsky, Mazin, and Vazirani s Parking Functions 5 Combinatorial Connection

3 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

4 Affine Springer Fiber Notation K := C((ɛ)) and O = C[[ɛ]] Fix n 1 V := K n be a K-vector space G := GL n (K) (alternatively SL n (K)) P := GL n (O) (alternatively SL n (O)) B := {g P g evalulated at ɛ = 0 is uppertriangular} (Iwahori subgroup)

5 Affine Springer Fiber Interpretation Consider elements of V as infinite C tuples a i ɛ i b i ɛ i c i ɛ i i N 1 i N 2 i N 3 (..., a 1, b 1, c 1, a 0, b 0, c 0, a 1, b 1, c 1,... ) ɛ acts by shifting n spots If {e i } n i=1 is a C basis of Cn, then e i+kn = ɛ k e i is a C basis of V

6 Affine Springer Fiber Affine Grassmannian Definition ([1, 4]) The affine Grassmannian G n for the group GL n 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 ɛ N O n M ɛ N O n (d) dim C ɛ N O n /M = dim C M/ɛ N O n Note G n = G/P.

7 Affine Springer Fiber Affine Flag Definition The affine flag ind-variety F n is the space of collections M 0 M 1 M n = ɛm 0 such that 1 For all i, M i satisfies (a), (b), and (c) for G n 2 M 0 G n 3 dim C M i /M i+1 = 1 Note F n = G/B

8 Affine Springer Fiber Affine Springer Fiber Consider the single shift of the C basis of V N = ɛ Let m > n, gcd(m, n) = 1 and m = nk + b T = N m and I n + T G Definition Let T be a semiregular nil-elliptic endomorphism. The Affine Springer Fiber is the set of fixed points of T over F n. Equivalently, if u = I n + T, then the Affine Springer Fiber is F u = F m/n.

9 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

10 Affine Symmetric Group Affine Symmetric Group Let E R n be perpendicular to (1, 1,..., 1) Let α i be the ith simple coroot (0, 0,..., 1, 1..., 0) with 1 in the ith position, 1 i < n Let H i E be the hyperplane perpendicular to α i {H i } acts on E by reflection Define s i to be the reflection across H i The complement of {H i }, closed under reflections, form Weyl chambers The fundamental chamber is on the positive side of all H i Assigning 1 S n to the fundamental chamber creates an isomorphism to S n

11 Affine Symmetric Group Example n = 3 H 1 H 2 α 2 1 α 1 s 1 s 2 s 1 s 2 s 2 s 1 s 2 s 1 s 2

12 Affine Symmetric Group Affine Symmetric Group (n = 3) s 0 s 1 s 2 s 1 s 0 s 1 s 2 s 0 s 2 s 1 s 1 s 0 s 1 s 2 s 0 s 1 s 0 s 2 s 1 s 0 s 1 s 2 s 1 s 1 s 0 s 1 s 2 s 0 s 2 s 1 s 0 s 2 s 0 s 2 s 2 s 0 s 2 s 1 s 2 H 1 0 s 1 s 0 s 2 s 1 s 1 s 0 s 2 s 1 s 0 s 1 1 s 2 s 2 s 0 s 2 s 0 s 1 s 2 s 0 s 1 s 2 s 1 s 2 s 2 s 1 s 2 s 1 s 2

13 Affine Symmetric Group Affine Symmetric Group Definition (Affine Symmetric Group) The affine symmetric group S n is the group of words in {s 0,..., s n 1 } with the following relations: s 2 i = 1 s i s j s i = s j s i s j if i j ±1 (mod n) s i s j = s j s i if i j ±1 (mod n) and n > 2

14 Affine Symmetric Group Affine Grassmannians Consider the cosets of S n /S n Definition ω S n is affine grassmannian if it is the minimum length representative of ωs n in S n /S n. Q := {(x 1,..., x n ) Z n n i=1 x i = 0} S n = Q S n If ω S n is affine grassmannian, then it is the identity or all its reduced words end with s 0

15 Affine Symmetric Group

16 Affine Symmetric Group

17 Affine Symmetric Group The Permutation Consider ω S n as a permutation Z Z 1 id Z We need only define the s i action ω(x + 1) if x i (mod n) (ωs i )(x) = ω(x 1) if x i + 1 (mod n) ω(x) otherwise n ω(i) = i=1 n(n + 1) 2 ω(x + n) = ω(x) + n Denoted by window notation [ω(1), ω(2),..., ω(n)]

18 Affine Symmetric Group Indexing Sets Recall F n = G/B and G n = G/P F n = BwB/B ω S n G n = BwP/P ω S n/s n

19 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

20 Hikita s Representation Notation F λ m/n F m/n F n π 1 C λ G m/n π G n π π : F n G n the natural projection (G m/n := π(f m/n )) G n has paving by Iwahori orbits BωP/P where ω S n /S n G m/n has a paving by G m/n (BωP/P) The non-zero cells are indexed by partitions, called C λ Fm/n λ = π 1 (C λ )

21 Hikita s Representation Hikita Representation P = {(a i ) a i Z 0, a n = 0, a 1,..., a n 1 0, a i+n = a i + 1} ˇλ : P Q is a bijection a = n 1 i=1 a i s a+l ˇλ(a 1,..., a n 1 ) = ˇλ(a 1,..., a l+1, a l,..., a n 1 ) if l = 1, 2,..., n 2, ˇλ(a n 1 1, a 1,..., a n 2 ) if l = n 1 and a n 1 1, ˇλ(a 1,..., a n 1 ) if l = n 1 and a n 1 = 0, ˇλ(a 2,..., a n 1, a 1 + 1) if l = 0.

22 Hikita s Representation Example (2,1) (2,0) (1,2) (0,2) (1,1) (0,1) (0,0) (1,0) (0,3) (3,0)

23 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 δ is the partition below the diagonal (0, n) to (m, 0) Proposition ([2]) There is a bijection {(a i ) P Cˇλ(ai ) } {λ λ is a partition, λ δ δ } (a i ) λ(a i ) := δ (a b, a 2b,..., a nb )

24 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

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

26 Gorsky, Mazin, and Vazirani s Parking Functions Parking Function Properties f ([n]), when sorted as a 1, a 2,..., a n, obeys a i i Any permutation of a parking function is a parking function There are (n + 1) n 1 parking functions on domain [n]

27 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 PF m/n = m n 1 Consider the parking function PF 5/4 but 2040 PF 7/4

28 Gorsky, Mazin, and Vazirani s Parking Functions Mapping m Sn S n is the set of m-restricted permutations m Sn = {ω i < j ω(j) ω(i) m} GMV created a map SP : m Sn PF m/n ω SP ω Proven to be a bijection for m = kn ± 1 SP is conjectured to be a bijection for all m SP ω (i) = #{j > i 0 < ω(i) ω(j) < m}

29 Gorsky, Mazin, and Vazirani s Parking Functions Example SP ω (i) = {j > i ω(i) m < ω(j) < ω(i)} n = 4, m = 7, ω = [4, 2, 3, 5] SP ω (1) = 3 SP ω (2) = 0 SP ω (3) = 1 SP ω (4) = 1 SP ω = 3011

30 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 F m/n F n admits an affine paving by m n 1 affine cells.

31 Gorsky, Mazin, and Vazirani s Parking Functions Theorem (GMV [1]) There is a natural bijection between the affine cells in F m/n and the affine permutations in m Sn. The dimension of the cell Σ ω labeled by the affine permutation ω is equal to n SP ω (i). i=1

32 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

33 Combinatorial Connection What do we know? F m/n is paved by m Sn G m/n is paved by a subset of S n /S n m Sn bijects with PF m/n Non-zero S n /S n bijects with P π F m/n Gm/n How to map from PF m/n P?

34 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) (a i ) = 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)

35 Combinatorial Connection Extended Example Convert (a i ) to S n (0,0) (0,1) (1,0) (1,1) (2,1) 1 s 0 s 2 s 0 s 1 s 0 s 2 s 1 s 2 s 0 (0,1) (1,1) (0,0) (2,1) (1,0) (1,1) (1,0) (0,1) (2,1) (0,0)

36 Combinatorial Connection Useful Facts GMV extended the H k i notation H k i,j = { x E x i x j = k} H k i H k i,i+1 for 1 i < n H k 0 Hk 1,n Hi,j k = H k j,i = Hi+tn,j+tn k = Hk 1 i,j n Dn m is the Sommers region bounded by {Hi,i+m 0 1 i n} Lemma (GMV) The set of alcoves {ω(a 0 ) ω m Sn } coincides with the set of alcoves that fit inside the region D m n.

37 Combinatorial Connection Extended Example Label D 4 3 Convert to windows Apply SP s 1 s 0 s 1 s 2 s 2 s 0 s 2 s [150] 101 [204] s 0 s [042] s 0 s 2s2 201 [4-13] s 102 [105] 1 s 0 s [024] s [-143] s 0 s [015] s 1 s [123] [-134] s 2 s [213] s [132] s [231] s 1 s [312] s 2 s 1 s 210 [321] 2 s 1 s 2 s [-226] s 1 s 2 s 0

38 Combinatorial Connection The Projection (0,1) (1,1) (0,0) (1,0) (2,1)

39 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

40 Combinatorial Connection The Conjectured Natural Mapping GMV also created A : S m n PF m/n a bijection S m n A PF m/n PS ω ω 1 PF m/n SP m Sn D m n

41 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]

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