# Mathematics Colloquia and Seminars

### Generalized coinvariant algebras and their quantum analogs

Algebra & Discrete Mathematics

 Speaker: Brendon Rhoades, UC San Diego Related Webpage: http://www.math.ucsd.edu/~bprhoades/ Location: 2112 MSB Start time: Wed, Dec 13 2017, 4:10PM

For two positive integers $k \leq n$, we introduce a quotient $R_{n,k}$ of the ring
$\mathbb{Q}[x_1, \dots, x_n]$ of polynomials in $n$ variables.  The ring $R_{n,k}$
has the structure of a graded $S_n$-module.  When $k = n$, the ring $R_{n,k}$ reduces
to the classical {\em coinvariant algebra} attached to the symmetric group. Algebraic
properties of $R_{n,k}$ are controlled by combinatorial properties of ordered set
partitions of $\{1, 2, \dots, n\}$ with $k$ blocks.  We also present versions of
the $R_{n,k}$ quotient which carry actions of the 0-Hecke algebra $H_n(0)$ and
the generic-parameter Hecke algebra $H_n(q)$.  These quantum quotients are constructed
using a quantum deformation of the orbit harmonics method of Garsia and Procesi.
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Joint with Jim Haglund, Jia Huang, Travis Scrimshaw, and Mark Shimozono.  

Note the special day!