Search

Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformation \tau _b of the generating series of bipartite maps, which generalizes the partition function of \beta-ensembles of random matrices. The Matching-Jack conjecture suggests that the coefficients c^\lambda _{\mu,\nu } of the function \tau _b in the power-sum basis are non-negative integer polynomials in the deformation parameter b. Dołęga and Féray have proved in 2016 the "polynomiality" part in the Matching-Jack conjecture, namely that coefficients c^\lambda _{\mu,\nu } are in \mathbb {Q}[b]. In this paper, we prove the "integrality" part, i.e. that the coefficients c^\lambda _{\mu,\nu } are in \mathbb {Z}[b]. The proof is based on a recent work of the author that deduces the Matching-Jack conjecture for marginal sums \overline { c}^\lambda _{\mu,l} from an analog result for the b-conjecture, established in 2020 by Chapuy and Dołęga. A key step in the proof involves a new connection with the graded Farahat-Higman algebra. [ABSTRACT FROM AUTHOR]

Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation \mathcal {A}: \mathbb {R}[t] \to \mathbb {R}[t] defined by \mathcal {A}(t^n) = A_n(t), where A_n(t) denotes the n-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator \mathcal {A}, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials. [ABSTRACT FROM AUTHOR]

Zeta values in Tate algebras were introduced by Pellarin in 2012. They are generalizations of Carlitz zeta values and play an increasingly important role in function field arithmetic. In this paper we prove a conjecture of Pellarin on identities for these zeta values. The proof is based on arithmetic properties of Carlitz zeta values and an explicit formula for Bernoulli-type polynomials attached to zeta values in Tate algebras. [ABSTRACT FROM AUTHOR]

In this paper, we give a representation-theoretic interpretation of the specialization Ew◦λ(q,∞) of the nonsymmetric Macdonald polynomial Ew◦λ(q, t) at t = ∞ in terms of the Demazure submodule Vw◦- (λ) of the level-zero extremal weight module V (λ) over a quantum affine algebra of an arbitrary untwisted type. Here, λ is a dominant integral weight, and w◦ denotes the longest element in the finite Weyl group W. Also, for each x ∈ W, we obtain a combinatorial formula for the specialization Exλ(q,∞) at t = ∞ of the nonsymmetric Macdonald polynomial Exλ(q, t) and also a combinatorical formula for the graded character gch Vx- (λ) of the Demazure submodule Vx- (λ) of V (λ). Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape λ. [ABSTRACT FROM AUTHOR]

The Lambert W function, giving the solutions of a simple transcendental equation, has become a famous function and arises in many applications in combinatorics, physics, or population dyamics just to mention a few. In this paper we construct and study in great detail a generalization of the Lambert W which involves some special polynomials and even combinatorial aspects. [ABSTRACT FROM AUTHOR]

In an earlier paper, we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras. Here, we consider full quivers (as well as pseudo-quivers) as a combinatoric tool in order to describe PI-varieties of algebras. Each full quiver is naturally associated to a polynomial that encapsulates trace-like properties of the underlying algebra. [ABSTRACT FROM AUTHOR]

Let $A=\oplus_{g\in G}A_g$ be a $G$-graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component $A_e$ to that of the whole of $A$, in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where $A$ is finite dimensional and $A_e$ has polynomial growth. [ABSTRACT FROM AUTHOR]

In this paper, we investigate $*$-polynomial identities of minimal degree for the algebra of $n\times n$ matrices over a field, where $n<5$ and $*$ is the transpose involution. We first present some basic generators, and then proceed to show that all other minimal degree identities can be derived from those. [ABSTRACT FROM AUTHOR]