In this paper, the finite algorithm for strict Hurwitz invariance of a convex combination of polynomials (due to Bialas [5] and Fu and Barmish [6]) is extended to a strict Schur invariance test and to generalized stability tests. These tests have a main advantage over the “Edge Theorem” in [4], i.e., they are computationally efficient and exact. However, the Edge Theorem is more versatile since...

We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine gln. Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto. Mathem...

In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents. We classify exponent 2 up to monoidal equivalence. also modular braided It turns out that Gauss sum is complete invariant for 2. This result can be viewed as categorical analog Arf's theorem on non-degenerate quadratic forms over fields characteristic

Lattices of polynomial KP and BKP $\tau$-functions labelled by partitions, with the flow variables equated to finite power sums, as well associated multipair multipoint correlation functions are expressed via generalizations Jacobi's bialternant formula for Schur Nimmo's Pfaffian ratio $Q$-functions. These obtained applying Wick's theorem fermionic vacuum expectation value representations in wh...

For n<105 all coefficients of Fn(x) are ±1 or 0. For n = 10S, the coefficient 2 occurs for the first time. Denote by A w the greatest coefficient of Fn(x) (in absolute value). Schur proved that lim sup -4n= °°. Emma Lehmer proved that An>cn l,z for infinitely many n. In fact she proved that infinitely many such w's are of the form pqr with p, q, and r prime. In the present note we are going to ...

Consider representation theory associated to symmetric groups, or to Hecke algebras in type A, or to q-Schur algebras, or to finite general linear groups in non-describing characteristic. Rock blocks are certain combinatorially defined blocks appearing in such a representation theory, first observed by R. Rouquier. Rock blocks are much more symmetric than general blocks, and every block is deri...

Hilbert’s proof, apart from the determination of the best possible constant π csc(π/p), was published by Weyl [7]. The calculation of the constant, and the integral analogue of Hilbert’s double series theorem (for p = 2) are due to Schur [6]. The generalizations to other p′s of both the discrete and integral versions of this result were discovered later on by Hardy and Riesz and published by Ha...

We provide a short proof of the theorem that every real multivariate polynomial has symmetric determinantal representation, which was first proved in Helton et al. (2006) [4] . then an example using our approach and extend results from field R to arbitrary F different characteristic 2. The new we take is only based on elementary theory determinants, Schur complements, basic properties polynomia...

Schur-Szegö composition of two polynomials of degree less or equal than a given positive integer n introduces an interesting semigroup structure on polynomial spaces and is one of the basic tools in the analytic theory of polynomials, see [4]. In the present paper we add several (apparently) new aspects to the previously known properties of this operation. Namely, we show how it interacts with ...

We define a new family F̃w(X) of generating functions for w ∈ S̃n which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions such as their symmetry and conjecture certain positivity properties. As an application, we relate these functions to the k-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. ...