Hyperplane Arrangements & Diagonal Harmonics

Transcription

1 Hyperplane Arrangements & Diagonal Harmonics Drew Armstrong arxiv:

2 Coinvariants

3 Coinvariants Theorems (Newton-Chevalley-etc): Let S n act on S = C[x 1,..., x n ] by permuting variables.

4 Coinvariants Theorems (Newton-Chevalley-etc): Let S n act on S = C[x 1,..., x n ] by permuting variables. Then we have S S n = C[p 1,..., p n ] n where p k = x k i are the i=1 power sum symmetric polynomials.

5 Coinvariants Theorems (Newton-Chevalley-etc): The coinvariant ring R := S/(p 1,..., p n ) is isomorphic to the regular representation: R = Sn CS n

6 Coinvariants Theorems (Newton-Chevalley-etc): The coinvariant ring R := S/(p 1,..., p n ) is isomorphic to the regular representation: R = Sn CS n And it s graded, with Hilbert series i dim R i q i = n (1 + q + + q j 1 )=[n] q! j=1 the q--factorial

7 Coinvariants What s next? Let S n act on DS = C[x 1,..., x n,y 1,..., y n ] diagonally.

8 Coinvariants What s next? Let S n act on DS = C[x 1,..., x n,y 1,..., y n ] diagonally. (Weyl) Then the ring of diagonal invariants is generated by the polarized power sums p k,l = n i=1 NOT algebraically independent DS S n x k i yi l for k + l > 0

9 Coinvariants Hard Theorem (Haiman, 2001): The diagonal coinvariant ring DR := DS/(p k,l : k + l > 0) has dimension (n + 1) n 1

10 Coinvariants Hard Theorem (Haiman, 2001): The diagonal coinvariant ring DR := DS/(p k,l : k + l > 0) has dimension (n + 1) n 1 Ongoing Project: Describe the (bigraded) Hilbert/Frobenius series! New science of parking functions

11 Affine Permutations

12 Affine Permutations Bijections: π : Z Z Periodic : k Z, π(k + n) =π(k)+n Frame of Reference: π(1) + π(2) + + π(n) = ( ) n +1 2

13 Affine Permutations Bijections: π : Z Z Periodic : Frame of Reference: π(1) + π(2) + + π(n) = example k Z, π(k + n) =π(k)+n ( ) n +1 k π(k) The window notation : π = [0, 2, 4]

14 Affine Permutations Bijections: π : Z Z Periodic : Frame of Reference: π(1) + π(2) + + π(n) = example k Z, π(k + n) =π(k)+n ( ) n +1 k π(k) The window notation : π = [0, 2, 4] Also observe: π = ( 3, 2)(0, 1)(3, 4)(6, 7)

15 Affine Permutations Define affine transpositions: ((i, j)) := k Z(i + kn, j + kn)

16 Affine Permutations Define affine transpositions: ((i, j)) := k Z(i + kn, j + kn) Then we have: S n = ((1, 2)), ((2, 3)),..., ((n, n + 1)) affine symmetric group generated by affine adjacent transpositions

17 Affine Permutations (Lusztig, 1983) says it s a Weyl group transposition reflection in hyperplane ((1, 2)) x 1 x 2 =0 ((2, 3)) x 2 x 3 =0. ((n 1,n)) x n 1 x n =0 ((n, n + 1)) x 1 x n =1

18 Affine Permutations (Lusztig, 1983) says it s a Weyl group transposition reflection in hyperplane ((1, 2)) x 1 x 2 =0 ((2, 3)) x 2 x 3 =0. ((n 1,n)) x n 1 x n =0 ((n, n + 1)) x 1 x n =1 Abuse of notation: S n = ((1, 2)), ((2, 3)),..., ((n 1,n)) finite symmetric group

19 Picture of Affine S3 ((3, 4)) ((2, 3)) group elements = alcoves ((1, 2))

20 Two ways to think

21 Two ways to think Way 1. S n = S n S n = = = (finite symmetric group) X (minimal coset reps) (which cone are you in?) X (where in the cone?) (permute window notation) X (into increasing order)

22 Two ways to think Way 1. S n = S n S n = = = (finite symmetric group) X (minimal coset reps) (which cone are you in?) X (where in the cone?) (permute window notation) X (into increasing order) example [6, 3, 8, 1] = [3, 1, 4, 2] [ 3, 1, 8, 6]

23 Picture of Way ! " 123 " minimal coset rep 321! "

24 Picture of Way For Posterity: 231! " 321! " 123 " 213 Note (finite) ascent sets in window notation = = {1} = {2} = {1, 2} 312

25 What if we invert?! " "! "

26 Invert!

27 This is Way 2 to think.

28 This is Way 2 to think. S n = S n Q n = S n Qn = { semi-direct product with the root lattice } (r 1,..., r n ) Z n : i r i =0

29 This is Way 2 to think. S n = S n Q n = S n Qn = { semi-direct product with the root lattice } (r 1,..., r n ) Z n : i r i =0 In terms of window notation: [6, 3, 8, 1] = (2, 1, 4, 3) + 4 (1, 1, 1, 1) finite permutation + n times a root division with remainder

30 This is Way 2 to think. a copy of S 3 around each root vector Q 3

31 Now for Shi and Ish

32 Now for Shi and Ish Consider a special simplex Bounded by: x 1 x 2 = 1 x 2 x 3 = 1... x n 1 x n = 1 x 1 x n = 2

33 Now for Shi and Ish Consider a special simplex It s a dilation of the fundamental alcove by a factor of n +1 Hence it contains (n + 1) n 1 alcoves!

34 Now for Shi and Ish Cut it with the Shi arrangement Shi arrangement: x i x j =0, 1 for all 1 i<j n

35 Now for Shi and Ish And consider the distance enumerator (call it shi )

36 Now for Shi and Ish And consider the distance enumerator (call it shi ) 0 example q shi = 6 + 6q +3q 2 + q

37 Now for Shi and Ish Next define a statistic on the root lattice:

38 Now for Shi and Ish Next define a statistic on the root lattice: Given r =(r 1,..., r n ) Q n let j be maximal such that r j is minimal.

39 Now for Shi and Ish Next define a statistic on the root lattice: Given r =(r 1,..., r n ) Q n let j be maximal such that r j is minimal. Then ish(r) := j n(r j + 1)

40 Now for Shi and Ish Next define a statistic on the root lattice: Given r =(r 1,..., r n ) Q n let j be maximal such that r j is minimal. Then ish(r) := j n(r j + 1) ish(2, 2, 2, 2, 0) = 4 5 ( 2 + 1) = 9 here n =5, j =4, r j = 2

41 Now for Shi and Ish ish spirals

42 Now for Shi and Ish ish spirals. example t ish = 6 + 6t +3t 2 + t

43 Now for Shi and Ish The joint distribution: shi ish symmetry??

44 Now for Shi and Ish The joint distribution: Conjectures: shi ish symmetry?? Joint Symmetry: q shi t ish = t shi q ish

45 Now for Shi and Ish The joint distribution: Conjectures: shi ish symmetry?? Joint Symmetry: q shi t ish = t shi q ish In fact, we have q shi t ish = Hilbert series of DR diagonal coinvariants

46 Now for Shi and Ish Finally... to each alcove ascent set Asc we associate the (Gessel) Fundamental Quasisymmetric Function F Asc = z z i1 i2 z in i 1 i n j Asc i j <i j+1

47 Now for Shi and Ish Finally... to each alcove ascent set we associate the (Gessel) Fundamental Quasisymmetric Function F Asc = Asc z z i1 i2 z in i 1 i n j Asc i j <i j+1 = F = F {1} = F {2} = F {1,2}

48 Now for Shi and Ish Finally... to each alcove ascent set we associate the (Gessel) Fundamental Quasisymmetric Function F Asc = Asc z z i1 i2 z in i 1 i n j Asc i j <i j+1 = F = F {1} = F {2} = F {1,2} + = schur(3) = schur(2, 1) = schur(1, 1, 1) (trivial representation) (the other one) (sign representation)

49 Now for Shi and Ish

50 Now for Shi and Ish q shi t ish F Asc = the (q,t)--catalan

51 Now for Shi and Ish (Shuffle) Conjecture: is the Frobenius series of q shi t ish F Asc " DR

52 Now for Shi and Ish Theorem (me): My shuffle conjecture = The Shuffle Conjecture (HHLRU05).

53 Now for Shi and Ish Theorem (me): My shuffle conjecture = The Shuffle Conjecture (HHLRU05). That is, (at least) two natural maps to parking functions: dinv shi area area ish bounce Haglund-Haiman-Loehr statistics

54 Where does it come from?

55 Where does it come from? If you invert this picture...

56 Where does it come from?...you will get this picture.

57 Where does it come from?...you will get this picture. The Shi Arrangement

58 Where does it come from? If you invert this picture...

59 Where does it come from?...you will get this picture.

60 Where does it come from?...you will get this picture. The Ish Arrangement (please see Brendon s talk)

61 Thanks! Þakka þér fyrir