Related papers: Identities of the Kauffman Monoid $\mathcal{K}_3$

We study mixed identities for oligomorphic automorphism groups of countable relational structures. Our main result gives sufficient conditions for such a group to not admit a mixed identity without particular constants. We study numerous…

In this paper we explore the topological properties of self-replicating, 3-dimensional manifolds, which are modeled by idempotents in the (2+1)-cobordism category. We give a classification theorem for all such idempotents. Additionally, we…

We present a polynomial-time algorithm that determines, given some choice rule, whether there exists an obviously strategy-proof mechanism for that choice rule.

In this paper we consider polynomial representability of functions defined over $Z_{p^n}$, where $p$ is a prime and $n$ is a positive integer. Our aim is to provide an algorithmic characterization that (i) answers the decision problem: to…

For any homogeneous identity between $q$-minors, we provide an identity between $P,Q$-minors.

The invariant polytope algorithm was a breakthrough in the joint spectral radius computation, allowing to find the exact value of the joint spectral radius for most matrix families~\cite{GP2013,GP2016}. This algorithm found many…

Dodgson polynomials appear in Schwinger parametric Feynman integrals and are closely related to the well known Kirchhoff (or first Symanzik) polynomial. In this article a new combinatorial interpretation and a generalisation of Dodgson…

This paper presents both a proof method and a result. The proof method presented is particularly suitable for uniformly proving families of identities satisfied by a family of recursive sequences. To illustrate the method, we study the…

This is a survey of results obtained jointly with E. Aljadeff and published in Adv. Math. 218 (2008), 1453-1495. We explain how to set up a theory of polynomial identities for comodule algebras over a Hopf algebra, and concentrate on the…

We search an identity basis for the adjoint Lie algebra of the algebra $M_{1,1}(K)$ over a field, where $K$ is either the infinite generated Grassmann algebra $E$ or $E^1$, the variant of the algebra with $1$. In particular, we prove that…

We consider certain functional identities on the matrix algebra $M_n$ that are defined similarly as the trace identities, except that the "coefficients" are arbitrary polynomials, not necessarily those expressible by the traces. The main…

We show how certain suitably modified N-modular diagrams of integer partitions provide a nice combinatorial interpretation for the general term of Zeilberger's KOH identity. This identity is the reformulation of O'Hara's famous proof of the…

We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over $\mathbb{D}_{2^n}$. The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm…

We study the way in which the abstract structure of a small overlap monoid is reflected in, and may be algorithmically deduced from, a small overlap presentation. We show that every C(2) monoid admits an essentially canonical C(2)…

We obtain new combinatorial identities for integral values of binary Krawtchouk polynomials $K^{2m}_p(x)$, $0\le p\le 2m$, by computing the characters of the $p$-exterior representations on certain elements of order 2 of $\mathrm{SO}(2m)$.…

Motivated by Berkovich's nine $q$-binomial identities involving the Legendre symbol $(\frac{d}{3})$, we establish a unified form of $q$-binomial identities of this type through a combinatorial approach. This unified form includes…

We show that the Identity Problem is decidable in polynomial time for finitely generated sub-semigroups of the group $\mathsf{UT}(4, \mathbb{Z})$ of $4 \times 4$ unitriangular integer matrices. As a byproduct of our proof, we also show the…

Moments of the characteristic polynomial of a random matrix taken from any of the three ensembles, orthogonal, unitary or symplectic, are given either as a determinant or a pfaffian or as a sum of determinants. For gaussian ensembles…

Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's…

Let L be a restricted Lie superalgebra with its restricted enveloping algebra u(L) over a field F of characteristic p>2. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2\times 2 matrices over F. We…