Séminaire Lotharingien de Combinatoire, 93B.122 (2025), 12 pp.

Maria M. Gillespie

Higher Specht Polynomials for the Diagonal Action

Abstract. We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables x₁,...,x_n and y₁,...,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, 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.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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