Related papers: Non-Symmetric Hall-Littlewood Polynomials

The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In…

This paper is about a family of symmetric rational functions that form a one-parameter generalization of the classical Hall-Littlewood polynomials. We introduce two sets of (skew and non-skew) functions that are akin to P and Q…

Let $A$ be the path algebra of a Dynkin quiver $Q$ over a finite field, and $\mathscr{P}$ be the category of projective $A$-modules. Denote by $C^1(\mathscr{P})$ the category of 1-cyclic complexes over $\mathscr{P}$, and…

We relate the combinatorics of Hall-Littlewood polynomials to that of abelian $p$-groups with alternating or Hermitian perfect pairings. Our main result is an analogue of the classical relationship between the Hall algebra of abelian…

We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of…

We provide a combinatorial description of the coefficients appearing in the expansion of Hall-Littlewood polynomials in terms of monomial symmetric functions. We also give a Littlewood-Richardson rule for Hall-Littlewood polynomials. For…

We study the specialization of the type A nonsymmetric Macdonald polynomials at $t=0$ based on the combinatorial formula of Haglund, Haiman, and Loehr. We prove that this specialization expands nonnegatively into the fundamental slide…

We give a Hecke algebra derivation of Macdonald's expansion formula for Hall-Littlewood polynomials in terms of semistandard Young tableaux. This is accomplished by first obtaining a Hecke algebra lift of the expansion coefficients and then…

We construct an action of the Hecke algebra $H_n(q)$ on a quotient of the polynomial ring $F[x_1, \dots, x_n]$, where $F = \mathbb{Q}(q)$. The dimension of our quotient ring is the number of $k$-block ordered set partitions of $\{1, 2,…

We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$,…

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [S\'em. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald…

Littlewood polynomials are polynomials with each of their coefficients in $\{-1,1\}$. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro…

Demazure characters, also known as key polynomials, generalize the classical Schur polynomials. In particular, when all variables are set equal to $1$, these polynomials count the number of integer points in a certain class of…

The quadratic algebras Q_n are associated with pseudo-roots of noncommutative polynomials. We compute the Hilbert series of the algebras Q_n and of the dual quadratic algebras Q_n^!

The $N=2$ topological Yang-Mills and holomorphic Yang-Mills theories on simply connected compact K\"{a}hler surfaces with $p_g\geq 1$ are reexamined. The $N=2$ symmetry is clarified in terms of a Dolbeault model of the equivariant…

We introduce two families of non-commutative symmetric functions that have analogous properties to the Hall-Littlewood and Macdonald symmetric functions.

We use the comultiplication to prove that Hall polynomials exist for all finite and affine quivers. In the finite and cyclic cases, this approach provides a new and simple proof of the existence of Hall polynomials. In general, these…

We introduce the notion of "hypergeometric" polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = \lambda_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis…

We describe a categorification of the Double Affine Hecke Algebra (${\mathcal{H}\kern -.4em\mathcal{H}}$) associated with an affine Lie algebra $\widehat{\mathfrak{g}}$, including a categorification of the polynomial representation and…

A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in $\{ \pm 1\}$. We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood polynomials with square discriminant.…