Related papers: All Kronecker coefficients are reduced Kronecker c…
In 2021, Daqing Wan and Ping Xi studied the equivalence of the Lang-Trotter conjecture for CM elliptic curves and the Hardy-Littlewood conjecture for primes represented by a quadratic polynomial. Wan and Xi provided an alternative…
Translated into the language of representations of quivers, a challenge in matrix pencil theory is to find sufficient and necessary conditions for a Kronecker representation to be a subfactor of another Kronecker representation in terms of…
The Krohn-Rhodes Theorem proves that a finite semigroup divides a wreath product of groups and aperiodic semigroups. Krohn-Rhodes complexity equals the minimal number of groups that are needed. Determining an algorithm to compute complexity…
This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with…
It can be shown that it is possible to find a representation of Hecke algebras within Clifford algebras of multivectors. These Clifford algebras possess a unique gradation and a possibly non-symmetric bilinear form. Hecke algebra…
Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is…
For fixed compact connected Lie groups H \subseteq G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is…
In this paper, we give a counter-example, in the general case, Kronecker theorem will derive contradiction. Kronecker theorem be correct after removing some conditions.
The Hardy--Littlewood inequality for complex homogeneous polynomials asserts that given positive integers $m\geq2$ and $n\geq1$, if $P$ is a complex homogeneous polynomial of degree $m$ on $\ell_{p}^{n}$ with $2m\leq p\leq\infty$ given by…
A theorem of A.A. Brudno says that the Kolmogorov-Sinai entropy of a subshift X over $\mathbb{N}$ with respect to an ergodic measure $\mu$ equals the asymptotic Kolmogorov complexity of almost every word $\omega$ in X. The purpose of this…
A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible…
We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients in the special case of Kostka numbers.
A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product of two \S_r-irreducibles into irreducibles.…
The Kronecker product of two Schur functions $s_{\mu}$ and $s_{\nu}$, denoted by $s_{\mu}*s_{\nu}$, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the…
We show that for almost every frequency alpha \in \R \setminus \Q, for every C^omega potential v:\R/\Z \to R, and for almost every energy E the corresponding quasiperiodic Schrodinger cocycle is either reducible or nonuniformly hyperbolic.…
We prove that any spectrum is equivalent to the nonconnective K-theory of a stable $\infty$-category. We use these results to construct a stable $\infty$-category $\mathcal{C}$ with a bounded t-structure such that…
We prove a conjecture of Stembridge concerning stability of Kronecker coefficients that vastly generalizes Murnaghan's theorem. The main idea is to identify the sequences of Kronecker coefficients in question with Hilbert functions of…
We show that the shifted rank, or srank, of any partition $\lambda$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_{\lambda}$ function in terms of power sum symmetric functions. This gives…
We make progress towards understanding the structure of Littlewood-Richardson coefficients $g_{\lambda,\mu}^{\nu}$ for products of Jack symmetric functions. Building on recent results of the second author, we are able to prove new cases of…
The group $SL(2,\mathbb{C})$ of all complex $2\times 2$ matrices with determinant one is closely related to the group $\boldsymbol{\mathcal{L}}_{+}^\uparrow$ of real $4\times 4$ matrices representing the restricted Lorentz transformations.…