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Generalized coinvariant algebras and their quantum analogs

Algebra & Discrete Mathematics

Speaker: Brendon Rhoades, UC San Diego
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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.  
Joint with Jim Haglund, Jia Huang, Travis Scrimshaw, and Mark Shimozono.  

Note the special day!