International Journal of Algebra and Computation

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an ${\mathrm{\text{SU}}}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at $t=1$. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most $10$. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary ${\mathrm{\text{SL}}}_2$- or ${\mathrm{\text{SU}}}_2$-module as well as its first three Laurent coefficients.

Motivated by the formula of partial integration, we introduce the concept of $\mathrm{\Omega }$-Rota–Baxter algebras, which generalizes the Rota–Baxter algebras introduced by Baxter. We then construct free objects in the category of $\mathrm{\Omega }$-Rota–Baxter algebras, via a method of Gröbner–Shirshov bases.

Two finite words $u$ and $v$ are $k$-binomially equivalent if, for each word $x$ of length at most $k$, $x$ appears the same number of times as a subsequence (i.e., as a scattered subword) of both $u$ and $v$. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the $k$-binomial equivalence. We provide an algorithm generating the $2$-binomial equivalence class of a word. For $k\ge 2$ and alphabet of $3$ or more symbols, the language made of lexicographically least elements of every $k$-binomial equivalence class and the language of singletons, i.e., the words whose $k$-binomial equivalence class is restricted to a single element, are shown to be non-context-free. As a consequence of our discussions, we also prove that the submonoid generated by the generators of the free nil-$2$ group (also called free nilpotent group of class $2$) on $m$ generators is isomorphic to the quotient of the free monoid ${\{1,\dots ,m\}}^{\ast }$ by the $2$-binomial equivalence.

We extend, in significant ways, the theory of truncated boolean representable simplicial complexes introduced in 2015. This theory, which includes all matroids, represents the largest class of finite simplicial complexes for which combinatorial geometry can be meaningfully applied.

We propose a criterion for preserving the regularity of a formal language representation when passing from groups to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes to its finite index subgroups, and to show that there exists no left order on a finitely generated acylindrically hyperbolic group such that the corresponding positive cone is represented by a quasi-geodesic regular language. We also answer one of Navas’ questions by giving an example of an infinite family of groups which admit a positive cone that is generated by exactly $k$ generators, for every $k\ge 3$. As a special case of our construction, we obtain a finitely generated positive cone for $F_2\times \mathbb{Z}$.

We provide an analogue of Wedderburn’s factorization method for central polynomials with coefficients in an octonion division algebra, and present an algorithm for fully factoring polynomials of degree $n$ with $n$ conjugacy classes of roots, counting multiplicities.

This paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of this paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers, substantiated by numerical data, are presented. A remarkable aspect of the observed asymptotic behavior is the so-called asymptotic irrelevance of quasigroup identities, whereby cancelation resulting from quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers for long words in a large number of generators.

Sergei Ovsienko proved that the Gelfand–Tsetlin variety for ${𝔤𝔩}_n$ is equidimensional and the dimension of all irreducible components equals $n(n-1)/2$. This implies in particular the equidimensionality of the nilfiber of the (partial) Kostant–Wallach map. We generalize this result for the $k$-partial Kostant–Wallach map and prove that all its fibers are equidimensional of dimension $n^2-(k+1)n+k(k+1)/2$. Also, we study certain subvarieties of the Gelfand–Tsetlin variety and show their equidimensionality which gives a new proof of Ovsienko’s theorem for ${𝔤𝔩}_2,{𝔤𝔩}_3$ and ${𝔤𝔩}_4$.

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type $A$ Renner monoid) was computed earlier by Dieng et al. [2], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type $C$ and use them to obtain an explicit description of the character table for the type $C$ Renner monoid Hecke algebra.