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### 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. <br><br> Joint with Jim Haglund, Jia Huang, Travis Scrimshaw, and Mark Shimozono.

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