Remarks on the paper “Skew Pieri rules for Hall–Littlewood functions” by Konvalinka and Lauve

Abstract

Abstract. In a recent paper Konvalinka and Lauve proved several skew Pieri rules for Hall-Littlewood polynomials. In this note we show that q-analogues of these rules are encoded in a q-binomial theorem for Macdonald polynomials due to Lascoux and the author. The Konvalinka-Lauve formulas and their q-analoguesWe refer the reader to [14] for definitions concerning Hall-Littlewood and Macdonald polynomials.Let P λ/µ = P λ/µ (X; t) and Q λ/µ = Q λ/µ (X; t) be the skew Hall-Littlewood polynomials, e r = P (1 r ) the rth elementary symmetric function, h r the rth complete symmetric function and q r = Q (r) . Then the ordinary Pieri formulas for HallLittlewood polynomials are given by [14] P µ e r = λ vs λ/µ (t)P λ (1.1a)where the sums on the right are over partitions λ such that |λ| = |µ| + r. The Pieri coefficient vs λ/µ (t) is given by [14, p. 215, (3.2so that vs λ/µ (t) is zero unless µ ⊆ λ with λ− µ a vertical r-strip. Similarly, hs λ/µ (t) vanishes unless µ ⊆ λ with λ − µ a horizontal r-strip, in which case [14, p. 218, (3.10)]To express the skew Pieri formulas, Konvalinka and Lauve [8] (see also [7]) introduced a third Pieri coefficientwhere n(λ/µ) := i≥1. Note that sk λ/µ (t) = 0 if µ ⊆ λ.

The contributions of Stanley to the fabric of symmetric and quasisymmetric functions

“…Specifically, they give an expression for the product of the skew Hall-Littlewood polynomial P λ/µ times h r as a signed sum of skew Hall-Littlewood polynomials, and do the same with e r or q r := (1 − t)P r in place of h r . Finally, in [War13], Warnaar shows that q-analogues of these three results from [KL13] can be derived from a q-binomial theorem for Macdonald polynomials of Lascoux and himself [LW11].…”

Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs

“…Similarly, f λ/µ (n) is the number of down walks from λ to µ of length n in βY. For example, F (21), (2) (2) = −3, f (211),(1) (2) = 2. Enumerator of the down graph of µY is g λ/µ .…”

On the expansion of certain vector-valued characters of $U_q (\mathfrak{gl}_n)$ with respect to the Gelfand–Tsetlin basis

“…The functions φ λ/β (q, t), Ω β/μ (q, t) have interpretations in terms of skew Macdonald polynomials (in parameters q, t) with plethystic substitutions. In particular, we have [12]), and both these quantities have nice factorized forms.…”

“…We recall that, as mentioned in the introduction, there is a p-adic interpretation for coefficients sk λ/µ (t) and thus for c λ,µ (t). More precisely, sk λ/µ (t) = t n(λ)−n(µ) α λ (µ; t −1 ); where α λ (µ; p) is the number of subgroups of type µ in a finite abelian p-group of type λ, see [12] for example, and the references therein.…”

“…Here the coefficients sk λ/µ (t) are those studied in [12,7] in the context of Pieri rules. Note that when t = p −1 for an odd prime p,…”