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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]

Using the theory of Macdonald [ Symmetric functions and Hall polynomials , The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group \mathfrak {S}_n are given by plethystically transformed Macdonald polynomials specialized at q=t. We generalize this to restricted rational Cherednik algebras of wreath product groups C_\ell \wr \mathfrak {S}_n and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [ Combinatorics, symmetric functions, and Hilbert schemes , Int. Press, Somerville, MA, 2003, pp. 39–111]. [ABSTRACT FROM AUTHOR]

Let A be an associative algebra with involution * over a field of characteristic zero. A central *-polynomial of A is a polynomial in non-commutative variables that takes central values in A. Here we prove the existence of two limits called the central *-exponent and the proper central *-exponent that give a measure of the growth of the central *-polynomials and proper central *-polynomials, respectively. Moreover, we compare them with the PI-*-exponent of the 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]

Let F be a characteristic zero field, let G be a residually finite group, and let W be a G-prime and polynomial identity F-algebra. By constructing G-graded central polynomials for W, we prove the G-graded version of Posner's theorem. More precisely, if S denotes all nonzero degree e central elements of W, the algebra S-1W is G-graded simple and finite dimensional over its center. Furthermore, we show how to use this theorem in order to recapture a result of Aljadeff and Haile stating that two G-simple algebras of finite dimension are isomorphix if and only if their ideals of graded identities coincide. [ABSTRACT FROM AUTHOR]

We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual) εn! of the Fomin-Kirillov algebras εn; these algebras are connected N-graded and are defined for n ≥ 2. We establish that the algebra εn! is module finite over its cεnter (and thus satisfies a polynomial idεntity), is Noetherian, and has Gelfand-Kirillov dimεnsion ⌊n/2⌋ for each n ≥ 2. We also observe that εn! is not prime for n ≥ 3. By a result of Roos, εn is not Koszul for n ≥ 3, so neither is εn! for n ≥ 3. Nevertheless, we prove that εn! is Artin-Schelter (AS-)regular if and only if n = 2, and that εn! is both AS-Gorεnstein and AS-Cohεn-Macaulay if and only if n = 2, 3. We also show that the depth of εn! is ≤ 1 for each n ≥ 2, conjecture that we have equality, and show that this claim holds for n = 2, 3. Several other directions for further examination of εn! are suggested at the εnd of this article. [ABSTRACT FROM AUTHOR]

We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field k of characteristic 0. The first class consists of those Hopf k-algebras that are connected graded as algebras, and the second class are those Hopf k-algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel’fand–Kirillov dimension. It is shown that if the Hopf algebra H is a connected graded Hopf algebra of finite Gel’fand–Kirillov dimension n, then H is a noetherian domain which is Cohen–Macaulay, Artin–Schelter regular, and Auslander regular of global dimension n. It has S² = IdH, and it is Calabi–Yau. Detailed information is also provided about the Hilbert series of H. Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k-algebras of finite Gel’fand–Kirillov dimension n with connected coalgebra, the underlying coalgebra is shown to be Artin–Schelter regular of global dimension n. Both these classes of Hopf algebras share many features in common with enveloping algebras of finite-dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k-algebra H of Gel’fand–Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U(g) for any Lie algebra g. [ABSTRACT FROM AUTHOR]

We prove a slice theorem around closed leaves in a singular Riemannian foliation, and we use it to study the C∞-algebra of smooth basic functions, generalizing to the inhomogeneous setting a number of results by G. Schwarz. In particular, in the infinitesimal case we show that this algebra is generated by a finite number of polynomials. [ABSTRACT FROM AUTHOR]

KLR algebras of affine ADE types are known to be properly stratified if the characteristic of the ground field is greater than some explicit bound. Understanding the strata of this stratification reduces to semicuspidal cases, which split into real and imaginary subcases. Real semicuspidal strata are well understood. We show that the smallest imaginary stratum is Morita equivalent to Huerfano-Khovanov's zigzag algebra tensored with a polynomial algebra in one variable. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above. [ABSTRACT FROM AUTHOR]

Abstract. Let K be an algebraically closed field of characteristic zero, and let K := K(t) be the rational function field over K. For each d ≥ 2, we consider the unicritical polynomial fd(z) := zd + t ∈ K[z], and we ask the following question: If we fix α ∈ K and integers M ≥ 0, N ≥ 1, and d ≥ 2, does there exist a place p ∈ SpecK[t] such that, modulo p, the point α enters into an N-cycle after precisely M steps under iteration by fd? We answer this question completely, concluding that the answer is generally affirmative and explicitly giving all counterexamples. This extends previous work by the author in the case that α is a constant point. [ABSTRACT FROM AUTHOR]

We explore orthogonal polynomials on the unit circle whose Schur parameters are {cαn}∞n=1, where 0 < ∣α∣, ∣c∣ < 1. Specifically, we derive two different generating functions. The first can be represented explicitly in terms of sums of a q-hypergeometric type and used to derive explicitly the underlying orthogonal polynomials, while the second obeys a functional differential equation and can be used to determine the asymptotic behaviour of these polynomials. Extending these constructs to orthogonal polynomials of the second kind, we are able to construct the Carathéodory function and examine the underlying orthogonality measure. [ABSTRACT FROM AUTHOR]

Let K be a global field of characteristic different from 2 and u(x) ∈ K[x] be an irreducible polynomial of even degree 2g ≥ 6 whose Galois group over K is either the full symmetric group S2g or the alternating group A2g. We describe explicitly how to choose (infinitely many) pairs of distinct t1, t2 ∈ K such that the g-dimensional Jacobian of a hyperelliptic curve y² = (x−t1)(x−t2))u(x) has no nontrivial endomorphisms over an algebraic closure of K and has big monodromy. [ABSTRACT FROM AUTHOR]

Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module categories are equivalent to the subquotient categories of the BGG category O(gln) which show up in categorification of gl(1|1)-representations. We construct diagrammatically the graded cellular structure and the properly stratified structure of these algebras. [ABSTRACT FROM AUTHOR]

An integral self-affine tile is the solution of a set equation AT = ∪d∊D(T +d), where A is an n×n integer matrix and D is a finite subset of Zn. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∊ Qn×n. We define rational self-affine tiles as compact subsets of the open subring Rn ×Πp Kp of the adèle ring AK, where the factors of the (finite) product are certain p-adic completions of a number field K that is defined in terms of the characteristic polynomial of A. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tile with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with Rn ×Πp {0} ≃ Rn. Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of the digit set D, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. This enables us to gain new results for tilings associated with numeration systems. [ABSTRACT FROM AUTHOR]

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra Ah generated by elements x,y, which satisfy yx -- xy = h, where h ∈ F[x]. We investigate the family of algebras Ah as h ranges over all the polynomials in F[x]. When h ≠ 0, the algebras Ah are subalgebras of the Weyl algebra A1 and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of Ah over arbitrary fields F and describe the invariants in Ah under the automorphisms. We determine the center, normal elements, and height one prime ideals of Ah, localizations and Ore sets for Ah, and the Lie ideal [Ah, Ah]. We also show that Ah cannot be realized as a generalized Weyl algebra over F[x], except when h ∈ F. In two sequels to this work, we completely describe the irreducible modules and derivations of Ah over any field. [ABSTRACT FROM AUTHOR]

The classical Grace theorem states that every circular domain in the complex plane C containing the zeros of a polynomial p(z) contains a zero of any of its apolar polynomials. We say that a closed domain Ω ⊆ C* is a locus of p(z) if it contains a zero of any of its apolar polynomials and is the smallest such domain with respect to inclusion. In this work we establish several general properties of the loci and show, in particular, that the property of a set being a locus of a polynomial is preserved under a Möbius transformation. We pose the problem of finding a locus inside the smallest disk containing the roots of p(z) and solve it for polynomials of degree 3. Numerous examples are given. [ABSTRACT FROM AUTHOR]

Let R be a Noetherian normal domain. Call an R-algebra A "locally A...¹ in codimension-one" if RP ⊗R A is a polynomial ring in one variable over RP for every height-one prime ideal P in R. We shall describe a general structure for any faithfully flat R-algebra A which is locally A¹ in codimensionone and deduce results giving sufficient conditions for such an R-algebra to be a locally polynomial algebra. We also give a recipe for constructing R-algebras which are locally A¹ in codimension-one. When R is a normal affine spot (i.e., a normal local domain obtained by a localisation of an affine domain), we give criteria for a faithfully flat R-algebra A, which is locally A¹ in codimensionone, to be Krull and a further condition for A to be Noetherian. The results are used to construct intricate examples of faithfully flat R-algebras locally A¹ in codimension-one which are Noetherian normal but not finitely generated. [ABSTRACT FROM AUTHOR]

Let G = exp(g) be a connected, simply connected, nilpotent Lie group and let ω be a continuous symmetric weight on G with polynomial growth. We determine the structure of all the two-sided closed ideals of the weighted group algebra Lω1(G) which are attached to a flat co-adjoint orbit. [ABSTRACT FROM AUTHOR]

We solve a class of lifting problems involving approximate polynomial relations (soft polynomial relations). Various associated C*-algebras are therefore projective. The technical lemma we need is a new manifestation of Akemann and Pedersen's discovery of the norm adjusting power of quasi-central approximate units. A projective C*-algebra is the analog of an absolute retract. Thus we can say that various noncommutative semialgebraic sets turn out to be absolute retracts. In particular we show that a noncommutative absolute retract results from the intersection of the approximate locus of a noncommutative homogeneous polynomial with the noncommutative unit ball. By unit ball we are referring to the C*-algebra of the universal row contraction. We show that various alternative noncommutative unit balls are also projective. Sufficiently many C*-algebras are now known to be projective so that we are able to show that the cone over any separable C*-algebra is the inductive limit of C*-algebras that are projective. [ABSTRACT FROM AUTHOR]