## Mathematics## Seminar |

## Rocky Mountain Algebraic Combinatorics Seminar

### Upcoming Seminars Schedule

### Constructive Membership Tests in Some Infinite Matrix Groups

Alexander Hulpke

Colorado State University

We describe algorithms and heuristics that allow us to express arbitrary
elements of SL_{n}(*Z*) and Sp_{2n}(*Z*) as products of
generators in particular "standard" generating sets. For elements obtained
experimentally as random products, it produces product expressions whose
lengths are competitive with the input lengths.

### Subregular $J$-rings of Coxeter systems as quotients of path algebras

Tianjuan Xu

University of Colorado, Boulder

The asymptotic Hecke algebra, or J-ring, of a Coxeter system is an
associative algebra closely related to the Hecke algebra of the system. We
study a subalgebra J_{C} of J which has a natural basis indexed by the
rigid elements of the Coxeter group, where “rigid" means having a unique
reduced word. Exploiting the rigidity property, we show that J_{C} can be
realized as a certain quotient of the path algebra of the double quiver of
the Coxeter diagram of the system. This allows us to use quiver
representations to answer representation-theoretical questions about J_{C},
such as when J_{C} is semisimple, in terms of graph-theoretical properties
of the Coxeter diagram.

Weber 223

4–6 pm

Friday, Sep 27, 2019

(Refreshments in Weber 117, 3:30–4 pm)

Colorado State University

This is a joint Denver U / UC Boulder / UC Denver / U of Wyoming / CSU seminar that meets biweekly. Anyone interested is welcome to join us at a local restaurant for dinner after the talks.