Related papers: The Frobenius transform of a symmetric function

A function $f$ from an Abelian group $(A,+)$ to an Abelian group $(B,+)$ is $(n, m, S)$ zero-difference (ZD), if $S=\{\lambda_\alpha \mid \alpha \in A\setminus\{0\}\}$ where $n=|A|$, $m=|f(A)|$ and $\lambda_\alpha=|\{x \in A \mid…

We introduce a new notion of a homogeneous pair for a pseudo-Riemannian metric $g$ and a positive function $f$ on a manifold $M$ admitting a free $\mathbb{R}_{>0}$-action. There are many examples admitting this structure. For example, (a) a…

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $\left[ g_{ij}\right]_{i,j\in \mathbb{N}}$ with at most one 1 in each row and each column such that…

In this work we will show that if $F$ is a positive integer, then the set ${\mathrm{Arf}}(F)=\{S\mid S \mbox{ is an Arf numerical semigroup with Frobenius number } F\}$ verifies the following conditions: 1) $\Delta(F)=\{0,F+1,\rightarrow\}$…

The determinantal identities of Hamel and Goulden have recently been shown to apply to a tableau-based ninth variation of skew Schur functions. Here we extend this approach and its results to the analogous tableau-based ninth variation of…

By the Fourier transformations, any group-invariant functions over finite Abelian groups are transformed into group-invariant functions over the character groups. In this paper, we calculate matrix elements of this transformations under…

Let $G$ be a discrete group. Let $\lambda : G \to B(\ell_2(G),\ell_2(G))$ be the left regular representation. A function $\ph : G \to \comp$ is called a completely bounded multiplier (= Herz-Schur multiplier) if the transformation defined…

The stated ${\rm SL}_n$-skein algebra $\mathscr{S}_{\hat{q}}(\mathfrak{S})$ of a surface $\mathfrak{S}$ is a quantization of the ${\rm SL}_n$-character variety, and is spanned over $\mathbb{Z}[\hat{q}^{\pm 1}]$ by framed tangles in…

This is an overview of recent results on the use of 2D Toda $\tau$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an…

We here revisit Fourier analysis on the Heisenberg group H^d. Whereas, according to the standard definition, the Fourier transform of an integrable function f on H^d is a one parameter family of bounded operators on L 2 (R^d), we define (by…

We study the torus-equivariant homology $H_*^T(\mathrm{Gr}_G)$ of the affine Grassmannian $\mathrm{Gr}_G$, where $G=\mathrm{Sp}_{2n}(\mathbb{C})$ is the symplectic group. This homology admits a natural ring structure and a Schubert basis,…

An important problem in the field of graph signal processing is developing appropriate overcomplete dictionaries for signals defined on different families of graphs. The Cayley graph of the symmetric group has natural applications in ranked…

An alternating sign matrix is a square matrix with entries 1, 0 and -1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the…

Let $i: A\to R$ be a ring morphism, and $\chi: R\to A$ a right $R$-linear map with $\chi(\chi(r)s)=\chi(rs)$ and $\chi(1_R)=1_A$. If $R$ is a Frobenius $A$-ring, then we can define a trace map $\tr: A\to A^R$. If there exists an element of…

Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the "moments" F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a…

When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\geq 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $SL_2\mathbb{C}$-characters…

Convolution admits a natural formulation as a functional operation on matrices. Motivated by the functional and entrywise calculi, this leads to a framework in which convolution defines a matrix transform that preserves positivity. Within…

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if $g(t)=\sum_{k=0}^m a_kt^k$ is a polynomial of degree $m$ with non-negative coefficients, then, for all positive operators $A,\,B$ and…

Let $R=\mathbb{F}_p[x_1,\ldots,x_n]$ and let $\mathbf{F}$ be the ring of Frobenius operators over $R$. We introduce a notion of Bernstein dimension and multiplicity for the class of finitely generated $\mathbf{F}$-modules whose structure…

The involution Stanley symmetric functions $\hat{F}_y$ are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for…