Séminaire Lotharingien de Combinatoire, 93B.140 (2025), 12 pp.

Maria Gillespie, Eugene Gorsky and Sean T. Griffin

A Geometric Interpretation and Skewing Formula for the Delta Theorem

Abstract. We show that the symmetric function Δ'_{e_k-1}e_n appearing in the Delta Theorem can be obtained from the symmetric function in the integer-slope Rectangular Shuffle Theorem by applying a Schur skewing operator. This generalizes a formula by the first and third authors for the Delta Theorem at t=0, and follows from work of Blasiak, Haiman, Morse, Pun, and Seelinger. We also provide a combinatorial proof of this identity, giving a new proof of the Rise Delta Theorem from the Rectangular Shuffle Theorem.

We then introduce a variety Y_n,k, which we call the affine Δ-Springer fiber, generalizing the affine Springer fiber studied by Hikita, whose Borel-Moore homology has an S_n action and a bigrading that corresponds to the Delta Theorem symmetric function rev_q ωΔ'_{e_k-1}e_n under the Frobenius character map. To prove this, we first similarly provide a geometric interpretation for the Rectangular Shuffle Theorem, and then use a geometric skewing identity along with the skewing formula above to obtain our results on Y_n,k.

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

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