Related papers: A Proof of Solomon's Rule
We present some Markovian approaches to prove universality results for some functions on the symmetric group. Some of those statistics are already studied in [Kammoun, 2018, 2020] but not the general case. We prove, in particular, that the…
Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a…
This paper establishes new upper bounds for the sum of the $k$ largest eigenvalues of symmetric matrices. When applied to the adjacency matrix of a graph, our results improve upon a related bound due to Mohar {\bf [On the sum of k largest…
Let G be a simple complex algebraic group and let K be a reductive subgroup of G such that the coordinate ring of G/K is a multiplicity free G-module. We consider the G-algebra structure of C[G/K], and study the decomposition into…
Arrangement graphs were introduced for their connection to computational networks and have since generated considerable interest in the literature. In a pair of recent articles by Chen, Ghorbani and Wong, the eigenvalues for the adjacency…
It is known that Plotkin's reduction theorem is very important for his theory of universal algebraic geometry [arXiv:math. GM/0210187], [arXiv:math. GM/0210194]. It turns out that this theorem can be generalized to arbitrary categories…
In this paper, a class of combinatorial identities is proved. A method is used which is based on the following rule: counting elements of a given set in two ways and making equal the obtained results. This rule is known as "counting in two…
Several elementary properties of the symmetric group $S_n$ extend in a nice way to the full transformation monoid $M_n$ of all maps of the set $X:=\{1,2,3,\dots,n\}$ into itself. The group $S_n$ turns out to be in some sense the torsion…
We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is…
Any permutation in the finite symmetric group can be written as a product of simple transpositions $s_i = (i~i+1)$. For a fixed permutation $\sigma \in \mathfrak{S}_n$ the products of minimal length are called reduced decompositions or…
We study properties of arithmetic sets coming from multiplicative number theory and obtain applications in the theory of uniform distribution and ergodic theory. Our main theorem is a generalization of K\'atai's orthogonality criterion.…
Given a graded bialgebra $H$, we let $\Delta^{\left[ k\right] }:H\rightarrow H^{\otimes k}$ and $m^{\left[ k\right] }:H^{\otimes k}\rightarrow H$ be its iterated (co)multiplications for all $k\in\mathbb{N}$. For any $k$-tuple $\alpha=\left(…
We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let $A$ be an $n \times n$ symmetric matrix with entries in the polynomial ring…
The aim of the inverse Galois problem is to find extensions of a given field whose Galois group is isomorphic to a given group. In this article, we are interested in subgroups of GL(2,Z/nZ) where n is an integer. We know that, in general,…
The group algebra of the permutation group is spanned by a set of elements called projectors. The coordinates of permutations expanded in projectors are matrix elements of irreducible representations. The projectors of the permutation group…
We prove a recent conjecture of Blanco and Petersen (arXiv:1206.0803v2) about an expansion formula for inversions and excedances in the symmetric group.
The article focuses on four aspects related to the descent algebras of type $A$. They are modular idempotents, higher Lie powers, higher Lie modules and the right ideals of the symmetric group algebras generated by the Solomon's descent…
We find the branching laws for the classical pairs $\mathrm{GL}(m, \mathbb{C}) \subset \mathrm{GL}(n, \mathbb{C})$, $\mathrm{Sp}(2m, \mathbb{C}) \subset \mathrm{Sp}(2n, \mathbb{C})$, $\mathrm{SO}(q, \mathbb{C}) \subset \mathrm{SO}(p,…
Graphings serve as limit objects for bounded-degree graphs. We define the ``cycle matroid'' of a graphing as a submodular setfunction, with values in [0,1], which generalizes (up to normalization) the cycle matroid of finite graphs. We…
In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent $\omega$ of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans…