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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 and under the diagonal action of the symmetric group .  This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring 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.