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### Higher Specht bases under the diagonal action

**Algebra & Discrete Mathematics**

Speaker: | Maria Gillespie, Colorado State University |

Related Webpage: | https://mathematicalgemstones.com/maria/ |

Location: | 2112 MSB |

Start time: | Fri, Apr 26 2024, 11:00AM |

We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ under the diagonal action of the symmetric group $S_n$. This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring $R_n$ due to Ariki, Terasoma, and Yamada, which has the advantage of respecting the decomposition into irreducibles.As our main application of the general theory, we provide a higher Specht basis for the hook shape Garsia--Haiman modules. In the process, we obtain a new formula for their doubly graded Frobenius series in terms of new generalized cocharge statistics on tableaux.