On combinatorial models for q-Whittaker and modified Hall–Littlewood polynomials [HBNI Th257]

Abstract

This thesis is structured into two distinct parts. The first part addresses two special cases of a conjecture proposed by Ayyer, Mandelshtam, and Martin concerning a bijection between two different combinatorial models for modified Macdonald polynomials. This portion of the work is based on [41]. The second part studies three different monomial expansions of qq-Whittaker polynomials and their connection to the representation theory of the affine Lie algebra sln^sln​. This is based on joint work with Aritra Bhattacharya and S. Viswanath [4, 5]. The modified Macdonald polynomials H~λ(X;q,t)H λ​(X;q,t), defined by Garsia and Haiman [16], form an important class of symmetric functions in the variables xixi​, with coefficients from N[q,t]N[q,t], and are indexed by partitions λλ. Two important specializations of H~λH λ​ are the qq-Whittaker polynomial Wλ(X;t)Wλ​(X;t) and the modified Hall–Littlewood polynomial H~λ(X;t)H λ​(X;t), each capturing a different aspect of the structure. We first consider Wλ′(X;t)Wλ′​(X;t) and H~λ(X;t)H λ​(X;t), interpreting them as the coefficients of the highest and lowest powers of qq in H~λ(X;q,t)H λ​(X;q,t), respectively, where λ′λ′ denotes the conjugate of the partition λλ. Two combinatorial formulas for H~λ(X;q,t)H λ​(X;q,t) are explored: one introduced by Haglund, Haiman, and Loehr [18], involving the statistics inv and maj defined on fillings of Young diagrams; and another due to Ayyer, Mandelshtam, and Martin [2], which introduces a new statistic called quinv in place of inv. These two formulas yield two distinct combinatorial interpretations and parameterizing sets for both Wλ′(X;t)Wλ′​(X;t) and H~λ(X;t)H λ​(X;t). In both cases, we construct bijections between these sets that preserve the content and the major index. For each filling in the parameterizing sets for the qq-Whittaker (resp. modified Hall–Littlewood) polynomials, we identify the major index with the charge (resp. cocharge) of associated words. We then utilize two characterizations of the charge — due to Lascoux–Schützenberger and Killpatrick — to demonstrate that these bijections possess the desired structure-preserving properties. These results confirm the conjecture proposed in [2] in the specified special cases.

In the second part of the thesis, we study the qq-Whittaker polynomials Wλ(Xn;q)Wλ​(Xn​;q), which arise as the coefficients of the highest power of tt in H~λ(Xn;q,t)H λ​(Xn​;q,t), as well as from the specialization of the symmetric Macdonald polynomials Pλ(Xn;q,t)Pλ​(Xn​;q,t) at t=0t=0, where Xn=(x1,x2,…,xn)Xn​=(x1​,x2​,…,xn​). We examine various monomial expansions of these polynomials, including those described by fermionic formulas [27, (0.2), (0.3)], the inv statistic from Haglund–Haiman–Loehr [18], and the quinv statistic from Ayyer–Mandelshtam–Martin [2]. The central combinatorial structures underlying these expansions include partition overlaid patterns and column-strict fillings. The partition overlaid patterns are intimately tied to the representation theory of the affine Lie algebra sln^sln​, incorporating tools such as projections, branching maps, and direct limits — all of which align with the Chari–Loktev basis of local Weyl modules. We reinterpret these structures within the framework of column-strict fillings and establish their fundamental properties. Additionally, we construct weight-preserving bijections between the two models that respect the operations of projections, branching, and direct limits. Finally, we connect our findings to the colored lattice paths approach to qq-Whittaker polynomials, as developed by Wheeler, Borodin, Garbali, and others.