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). June 8, 2018 Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 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 . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 2 / 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. Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 2 / 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 ) } . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 2 / 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 June 8, 2018 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 . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 3 / 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 . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 3 / 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. Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 3 / 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 June 8, 2018 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 . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 4 / 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 } . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 4 / 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 . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 4 / 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 . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 4 / 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 June 8, 2018 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 the symmetric group S n (or more generally, any Coxeter group). Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 5 / 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 the symmetric group S n (or more generally, any Coxeter group). Norton (1979) studied the representation theory of H n (0) over an arbitrary field F . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 5 / 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 the symmetric group 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 . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 5 / 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 the symmetric group 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 June 8, 2018 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 ∈ S n } . Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 6 / 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 ∈ S n } . 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 Jia Huang (UNK) 0-Hecke algebra actions June 8, 2018 6 / 24