Hyperplane Arrangements & Diagonal Harmonics

Hyperplane Arrangements & Diagonal Harmonics Drew Armstrong arxiv:1005.1949

Coinvariants

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

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.

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

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

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

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

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

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

Affine Permutations

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

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 2 1 0 1 2 3 4 5 6 π(k) 3 1 1 0 2 4 3 5 7 2 The window notation : π = [0, 2, 4]

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

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

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

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

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

Two ways to think

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)

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]

Picture of Way 1 132 231! " 123 " minimal coset rep 321! " 213 312

Picture of Way 1 132 For Posterity: 231! " 321! " 123 " 213 Note (finite) ascent sets in window notation = = {1} = {2} = {1, 2} 312

What if we invert?! " "! "

Invert!

This is Way 2 to think.

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

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

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

Now for Shi and Ish

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

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!

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

Now for Shi and Ish And consider the distance enumerator (call it shi ) 0 1 0 1 1 2 3 2 1 0 0 1 2 1 0 0

Now for Shi and Ish And consider the distance enumerator (call it shi ) 0 example q shi = 6 + 6q +3q 2 + q 3 1 0 1 1 2 3 2 1 0 0 1 2 1 0 0

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

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.

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)

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

Now for Shi and Ish ish spirals. 3 1 0 0 0 0 1 1 1 1 0 0 1 2 2 2

Now for Shi and Ish ish spirals. example t ish = 6 + 6t +3t 2 + t 3 3 1 0 0 0 0 1 1 1 1 0 0 1 2 2 2

Now for Shi and Ish The joint distribution: shi 0 1 2 3 0 1 2 2 1 1 2 3 1 2 2 1 3 1 ish symmetry??

Now for Shi and Ish The joint distribution: Conjectures: shi 0 1 2 3 0 1 2 2 1 1 2 3 1 2 2 1 3 1 ish symmetry?? Joint Symmetry: q shi t ish = t shi q ish

Now for Shi and Ish The joint distribution: Conjectures: shi 0 1 2 3 0 1 2 2 1 1 2 3 1 2 2 1 3 1 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

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

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)

Now for Shi and Ish 01 01 11 11 10 20 30 21 11 00 20 02 03 10 12 02

Now for Shi and Ish q shi t ish F Asc = 01 01 1 + 11 11 10 20 30 21 11 + 1 1 1 1 1 + 03 00 10 20 12 02 02 1 1 1 1 1 the (q,t)--catalan

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

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

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

Where does it come from?

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

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

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

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

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

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

Thanks! Þakka þér fyrir