Related papers: $q$-Whittaker polynomials: bases, branching and di…
We consider the monomial expansion of the $q$-Whittaker polynomials given by the fermionic formula and via the inv and quinv statistics. We construct bijections between the parametrizing sets of these three models which preserve the $x$-…
We study inhomogeneous $q$-Whittaker polynomials which extend both $q$-Whittaker and stable Grothendieck polynomials. We prove that inhomogeneous $q$-Whittaker polynomials (in countably many variables) form a basis of certain commutative…
Exploiting the fact that the $q$-Whittaker polynomials arise as a specialization of the (modified) Macdonald polynomials, we derive some of their basic properties, and explore interesting identities that they satisfy. We also show how they…
We introduce and study a one-parameter generalization of the q-Whittaker symmetric functions. This is a family of multivariate symmetric polynomials, whose construction may be viewed as an application of the procedure of fusion from…
We introduce and study an inhomogeneous generalization of the spin $q$-Whittaker polynomials from [Borodin,Wheeler-17]. These are symmetric polynomials, and we prove a branching rule, skew dual and non-dual Cauchy identities, and an…
We introduce a new family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ arising from exactly solvable lattice models associated with the quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$. The…
The $(q,r)$-Whitney numbers were recently defined in terms of the $q$-Boson operators, and several combinatorial properties which appear to be $q$-analogues of similar properties were studied. In this paper, we obtain elementary and…
We consider the monomial expansion of the $q$-Whittaker and modified Hall-Littlewood polynomialsarising from specialization of the modified Macdonald polynomial. The two combinatorial formulas for the latter due to Haglund, Haiman, and…
We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier type systems. We prove Ismail conjecture…
Bender, Coley, Robbins and Rumsey posed the problem of counting the number of subspaces which have a given profile with respect to a linear endomorphism defined on a finite vector space. Several special cases of this problem have been…
Three $q$-versions of Lommel polynomials are studied. Included are explicit representations, recurrences, continued fractions, and connections to associated Askey--Wilson polynomials. Combinatorial results are emphasized, including a…
Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general…
We discuss the problem posed by Bender, Coley, Robbins and Rumsey of enumerating the number of subspaces which have a given profile with respect to a linear operator over the finite field $\mathbb{F}_q$. We solve this problem in the case…
We will describe solvable lattice models whose partition functions depend on two sets of variables, $x_1,\cdots,x_n$ and $y_1, y_2, \cdots $ that have different connections with the representation theory of $\text{GL}(n,F)$ where $F$ is a…
We provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of the pin cover $\wti W$, a certain double cover of the Weyl group $W$, and an extended Dirac operator for graded Hecke…
The main purpose of this paper is to provide a novel approach to deriving formulas for the p-adic q-integral including the Volkenborn integral and the p-adic fermionic integral. By applying integral equations and these integral formulas to…
We use directed graphs called "syzygy quivers" to study the asymptotic growth rates of the dimensions of the syzygies of representations of finite dimensional algebras. For any finitely generated representation of a monomial algebra, we…
We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential…
We establish new properties of inhomogeneous spin $q$-Whittaker polynomials, which are symmetric polynomials generalizing $t=0$ Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come…
We study the spectral theory for the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are…