## Jae-ho Lee (University of Wisconsin at Madison)-Q-polynomial distance-regular graphs and the DAHA of rank one

Mathematics - Colloquium

Friday, November 9, 2012

2:00 PM-3:00 PM

Stratton Hall

308

Abstract: Let denote a Q-polynomial distance-regular graph with vertex set X. We assume that has q-Racah type and contains a Delsarte clique C. Fix a vertex xC. We partition X according to the path-length distance to both x and C. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a C-vector space W. The universal DAHA of type (C_1^V C_1 ) is the C-algebra H defined by generators {t_n^(±1) }_(n=0)^3 and relations (i) t_n t_n^(-1)=t_n^(-1) t_n=1 ; (ii) t_n+t_n^(-1) is central ; (iii) t_0 t_1 t_2 t_3=q^(-1/2).

We display an H-module structure for W. For this module and up to affine transformation,

(a) t_0 t_1+(t_0 t_1 )^(-1) acts as the adjacency matrix of ;

(b) t_3 t_0+(t_3 t_0 )^(-1) acts as the dual adjacency matrix of with respect to C;

(c) t_1 t_2+(t_1 t_2 )^(-1) acts as the dual adjacency matrix of with respect to x.

Suggested Audiences:
Adult, College

E-mail:
ma-chair@wpi.edu

Last Modified: November 6, 2012 at 10:58 AM

## Jae-ho Lee (University of Wisconsin at Madison)-Q-polynomial distance-regular graphs and the DAHA of rank one

Mathematics - Colloquium

Friday, November 9, 2012

2:00 PM-3:00 PM

Stratton Hall

308

Abstract: Let denote a Q-polynomial distance-regular graph with vertex set X. We assume that has q-Racah type and contains a Delsarte clique C. Fix a vertex xC. We partition X according to the path-length distance to both x and C. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a C-vector space W. The universal DAHA of type (C_1^V C_1 ) is the C-algebra H defined by generators {t_n^(±1) }_(n=0)^3 and relations (i) t_n t_n^(-1)=t_n^(-1) t_n=1 ; (ii) t_n+t_n^(-1) is central ; (iii) t_0 t_1 t_2 t_3=q^(-1/2).

We display an H-module structure for W. For this module and up to affine transformation,

(a) t_0 t_1+(t_0 t_1 )^(-1) acts as the adjacency matrix of ;

(b) t_3 t_0+(t_3 t_0 )^(-1) acts as the dual adjacency matrix of with respect to C;

(c) t_1 t_2+(t_1 t_2 )^(-1) acts as the dual adjacency matrix of with respect to x.

Suggested Audiences: Adult, College

E-mail: ma-chair@wpi.edu

Last Modified: November 6, 2012 at 10:58 AM