0-Hecke algebra actions on quotients of polynomial rings Jia Huang - PowerPoint PPT Presentation

0-Hecke algebra actions on quotients of polynomial rings Jia Huang University of Nebraska at Kearney E-mail address : huangj2@unk.edu Part of this work is joint with Brendon Rhoades (UCSD). December 28, 2017 Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 1 / 24

The Symmetric Group S n The symmetric group S n := { bijections on { 1 , . . . , n }} is generated by the adjacent transpositions s i = ( i , i + 1), 1 ≤ i ≤ n − 1, with quadratic relations s 2 i = 1, 1 ≤ i ≤ n − 1, and braid relations � s i s i +1 s i = s i +1 s i s i +1 , 1 ≤ i ≤ n − 2 , s i s j = s j s i , | i − j | > 1 . More generally, a Coxeter group has a similar presentation. The length of any w ∈ S n is ℓ ( w ) := min { k : w = s i 1 · · · s i k } , which coincides with inv ( w ) := { ( i , j ) : 1 ≤ i < j ≤ n , w ( i ) > w ( j ) } . For example, w = 3241 ∈ S 4 has ℓ ( w ) = inv ( w ) = 4 and reduced repressions w = s 2 s 1 s 2 s 3 = s 1 s 2 s 1 s 3 = s 1 s 2 s 3 s 1 . Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 2 / 24

The Hecke Algebra H n ( q ) The (Iwahori-)Hecke algebra H n ( q ) is a deformation of the group algebra F S n of S n over an arbitrary field F . It is an F ( q )-algebra generated by T 1 , . . . , T n − 1 with relations   ( T i + 1)( T i − q ) = 0 , 1 ≤ i ≤ n − 1 ,  T i T i +1 T i = T i +1 T i T i +1 , 1 ≤ i ≤ n − 2 ,   T i T j = T j T i , | i − j | > 1 . It has an F ( q )-basis { T w : w ∈ S n } , where T w := T s 1 · · · T s k if w = s 1 · · · s k with k minimum. It has significance in algebraic combinatorics, knot theory, quantum groups, representation theory of p-adic groups, etc. Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 3 / 24

The 0-Hecke algebra H n (0) Set q = 1: H n ( q ) → F S n , T i → s i , T w → w . Tits showed that H n ( q ) ∼ = C S n unless q ∈ { 0 , roots of unity } . Set q = 0: H n ( q ) → H n (0), T i → π i , T w → π w ,  π 2  i = − π i , 1 ≤ i ≤ n − 1 ,  π i π i +1 π i = π i +1 π i π i +1 , 1 ≤ i ≤ n − 2 ,   π i π j = π j π i , | i − j | > 1 . H n (0) has another generating set { π i := π i + 1 } , with relations  π 2  i = π i , 1 ≤ i ≤ n − 1 ,  π i π i +1 π i = π i +1 π i π i +1 , 1 ≤ i ≤ n − 2 ,   π i π j = π j π i , | i − j | > 1 . Sending π i to − π i gives an algebra automorphism. Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 4 / 24

Significance of the 0-Hecke algebra Using the automorphism π i �→ − π i of H n (0), Stembridge (2007) gave a short derivation for the M¨ obius function of the Bruhat order of S n (or more generally, any Coxeter group). Norton (1979) studied the representation theory of H n (0) over an arbitrary field F . Norton’s result provides motivations to work of Denton, Hivert, Schilling, and Thi´ ery (2011) on the representation theory of finite J -trivial monoids . Krob and Thibon (1997) discovered connections between H n (0)-representations and certain generalizations of symmetric functions, which is similar to the classical Frobenius correspondence between S n -representations and symmetric functions. Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 5 / 24

Analogies between S n and H n (0) F S n is the group algebra of the symmetric group S n and H n (0) is the monoid algebra of the monoid { π w : w ∈ W } . The defining representations of S n and H n (0) are analogous: s n − 1 s 1 s 2 � 2 � � · · · � � n 1 � π n − 1 π 1 π 2 � 2 � · · · � n 1 S n acts on Z n : s i swaps a i and a i +1 in a 1 · · · a n . H n (0) acts on Z n by the bubble-sorting operators : π i swaps a i and a i +1 in a 1 · · · a n if a i > a i +1 , or fixes a 1 · · · a n otherwise. Analogies between other representations of S n and H n (0)? Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 6 / 24

Actions on polynomials S n acts on F [ X ] := F [ x 1 , . . . , x n ] by variable permutation. H n (0) also acts on F [ X ] via the Demazure operators π i ( f ) := ∂ i ( x i f ) = x i f − s i ( x i f ) . x i − x i +1 The divided difference operator ∂ i is useful in Schubert calculus, a branch of algebraic geometry. π 1 ( x 3 1 x 2 x 3 x 4 4 ) = ( x 3 1 x 2 + x 2 1 x 2 2 + x 1 x 3 2 ) x 3 x 4 4 . π 2 ( x 3 1 x 2 x 3 x 4 4 ) = x 3 1 x 2 x 3 x 4 4 . π 3 ( x 3 1 x 2 x 3 x 4 4 ) = x 3 1 x 2 ( − x 2 3 x 3 4 − x 3 3 x 2 4 ). Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 7 / 24

The coinvariant algebra of S n The invariant ring F [ X ] S n := { f ∈ F [ X ] : wf = f , ∀ w ∈ S n } consists of all symmetric functions in x 1 , . . . , x n . It is a polynomial ring F [ X ] S n = F [ e 1 , . . . , e n ] in the elementary symmetric functions � e k := x i 1 · · · x i k , k = 1 , . . . , n . 1 ≤ i 1 < ··· < i k ≤ n n = 3: e 1 = x 1 + x 2 + x 3 , e 2 = x 1 x 2 + x 1 x 3 + x 2 x 3 , e 3 = x 1 x 2 x 3 If f ∈ F [ X ] S n and g ∈ F [ X ], then s i ( fg ) = fs i ( g ). Thus F [ X ] / ( e 1 , . . . , e n ) becomes a graded S n -module. Theorem (Chevalley–Shephard–Tod 1955, indirect proof) The coinvariant algebra F [ X ] / ( e 1 , . . . , e n ) is isomorphic to the regular representation F S n of S n , if F is a field of characteristic 0 . Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 8 / 24

The coinvariant algebra of H n (0) The H n (0)-invariants are also the symmetric functions: π i f = f if and only if s i f = f for all i . If f ∈ F [ X ] S n and g ∈ F [ X ], then π i ( fg ) = f π i ( g ). Thus F [ X ] / ( e 1 , . . . , e n ) becomes a graded H n (0)-module. Theorem (H. 2014) The coinvariant algebra F [ X ] / ( e 1 , . . . , e n ) is isomorphic to the regular representation of H n (0) . Remark Our proof is constructive, using the descent basis of the coinvariant algebra given by Garsia and Stanton (1984). Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 9 / 24

� � � � � � � � � � � � H 3 (0) ∼ = F [ x 1 , x 2 , x 3 ] / ( e 1 , e 2 , e 3 ) π 1 = π 2 =0 π 1 = π 2 =0 ch t 1 2 3 1 s 3 π 1 = − 1 π 1 =0 ,π 2 = − 1 π 1 = − 1 π 1 =0 ,π 2 = − 1 + 1 3 1 2 π 2 − − − → 2 3 π 2 x 2 x 3 t s 12 − − − → + 2 1 π 1 − − − → t 2 s 21 π 1 1 3 2 3 x 1 x 3 x 2 x 3 − − − → π 2 = − 1 π 1 = − 1 ,π 2 =0 + π 2 = − 1 π 1 = − 1 ,π 2 =0 1 t 3 s 111 x 2 x 2 2 π 1 = π 2 = − 1 π 1 = π 2 = − 1 3 3 Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 10 / 24

Representation theory of S n Every S n -module is a direct sum of simple modules. A partition of n is a decreasing sequence λ = ( λ 1 , . . . , λ k ) of positive integers whose sum is n ; this is denoted by λ ⊢ n . The simple S n -modules S λ are indexed by partitions λ ⊢ n . The Schur function s λ is the sum of x τ for all semistandard tableaux τ of shape λ . For example, + · · · = x 2 1 x 2 + x 1 x 2 s 21 = x 1 1 + x 1 2 2 + · · · . 2 2 Symmetric functions form a graded Hopf algebra with a self-dual basis { s λ } . The Frobenius characteristic map S λ �→ s λ is an isomorphism from S n -representations to Sym . Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 11 / 24

Representation theory of H n (0) A composition of n , denoted by α | = n , is a sequence α = ( α 1 , . . . , α ℓ ) of positive integers whose sum is n . Norton (1979) showed that H n (0) = � = n P α , so every projective α | indecomposable H n (0)-module is isomorphic to P α for some α | = n . Furthermore, every simple H n (0)-module is isomorphic to some C α := top ( P α ) = P α / rad P α , which is 1-dimensional. Generalizing Sym are two graded Hopf algebras QSym ( quasisymmetric functions ) and NSym ( noncommutative symmetric functions ) with dual bases { F α } and { s α } . Krob and Thibon (1997): by P α �→ s α and C α �→ F α one has { H n (0)-modules } ↔ QSym (up to composition factors), { projective H n (0)-modules } ↔ NSym . Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 12 / 24

� � � � � � � � � � � � H 3 (0) ∼ = F [ x 1 , x 2 , x 3 ] / ( e 1 , e 2 , e 3 ) π 1 = π 2 =0 π 1 = π 2 =0 ch t 1 2 3 1 s 3 π 1 = − 1 π 1 =0 ,π 2 = − 1 π 1 = − 1 π 1 =0 ,π 2 = − 1 + 1 3 1 2 π 2 − − − → 2 3 π 2 x 2 x 3 t s 12 − − − → + 2 1 π 1 − − − → t 2 s 21 π 1 1 3 2 3 x 1 x 3 x 2 x 3 − − − → π 2 = − 1 π 1 = − 1 ,π 2 =0 + π 2 = − 1 π 1 = − 1 ,π 2 =0 1 t 3 s 111 x 2 x 2 2 π 1 = π 2 = − 1 π 1 = π 2 = − 1 3 3 Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 13 / 24

� � � � � � � � � � � � � � � α = (1 , 2 , 1) x 2 · x 2 x 1 x 4 = x 1 x 2 3 2 x 4 1 4 2 π 1 = π 3 = − 1 π 2 π 2 2 x 1 x 2 3 x 4 + x 1 x 2 x 3 x 4 1 4 3 π 2 = − 1 π 1 π 3 1 2 π 1 π 3 x 2 x 2 x 1 x 3 x 2 2 4 1 3 3 x 4 4 3 4 π 3 π 1 π 1 = π 2 = − 1 π 2 = π 3 = − 1 π 3 π 1 1 2 3 x 2 x 3 x 2 4 4 π 1 = π 3 = − 1 ,π 2 =0 Jia Huang (UNK) 0-Hecke algebra actions December 28, 2017 14 / 24