Representations Of Monoids Research Articles

Split Grothendieck rings of rooted trees and skew shapes via monoid representations

We study commutative ring structures on the integral span of rooted trees and $n$-dimensional skew shapes. The multiplication in these rings arises from the smash product operation on monoid representations in pointed sets. We interpret these as Grothendieck rings of indecomposable monoid representations over $\fun$ - the "field" of one element. We also study the base-change homomorphism from $\mt$-modules to $k[t]$-modules for a field $k$ containing all roots of unity, and interpret the result in terms of Jordan decompositions of adjacency matrices of certain graphs.

The Hopf algebra of skew shapes, torsion sheaves on [formula omitted], and ideals in Hall algebras of monoid representations

We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the representation possess a compatible grading, and conditions on the support of the module. Quotients by these ideals lead to combinatorial Hopf algebras which can be interpreted as Hall algebras of certain sub-categories of modules. In the case of the free commutative monoid on n generators, we obtain a co-commutative Hopf algebra structure on n-dimensional skew shapes, whose underlying associative product amounts to a “stacking” operation on the skew shapes. The primitive elements of this Hopf algebra correspond to connected skew shapes, and form a graded Lie algebra by anti-symmetrizing the associative product. We interpret this Hopf algebra as the Hall algebra of a certain category of coherent torsion sheaves on A/F1n supported at the origin, where F1 denotes the field of one element. This Hopf algebra may be viewed as an n-dimensional generalization of the Hopf algebra of symmetric functions, which corresponds to the case n=1.

Knuth’s Coherent Presentations of Plactic Monoids of Type A

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth’s generators.

COMPLETELY REDUCIBLE SETS

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.

Representation Theory over Tropical Semifield and Langlands Duality

Recently we propose a class of infinite-dimensional integral representations of classical \({\mathfrak{gl}_{\ell+1}}\)-Whittaker functions and local Archimedean local L-factors using two-dimensional topological field theory framework. The local Archimedean Langlands duality was identified in this setting with the mirror symmetry of the underlying topological field theories. In this note we introduce elementary analogs of the Whittaker functions and the Archimedean L-factors given by Ul+1-equivariant symplectic volumes of appropriate Kahler Ul+1-manifolds. We demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids \({GL_{\ell+1}(\mathcal{R}),\; \mathcal{R}}\) being the tropical semifield. We also show that the elementary Whittaker functions can be obtained from the non-Archimedean Whittaker functions over \({\mathbb{Q}_p}\) by taking the formal limit p→ 1. Hence the elementary special functions constructed in this way might be considered as functions over the mysterious field \({\mathbb{Q}_1}\). The existence of two representations for the elementary Whittaker functions, one as an equivariant volume and the other as a matrix element, should be considered as a manifestation of a hypothetical elementary analog of the local Langlands duality for number fields. We would like to note that the elementary local L-factors coincide with L-factors introduced previously by Kurokawa.

Homomorphism-Homogeneous Relational Structures

We study relational structures (especially graphs and posets) which satisfy the analogue of homogeneity but for homomorphisms rather than isomorphisms. The picture is rather different. Our main results are partial characterizations of countable graphs and posets with this property; an analogue of Fraisse's theorem; and representations of monoids as endomorphism monoids of such structures.

Representations of monoids arising from finite groups of Lie type

A class of finite monoids M M constructed from a group G G of Lie type is considered. We describe the irreducible complex representations and prove the complete reducibility of the representations of M M . The sandwich matrix of M M is decomposed into a product of matrices corresponding to maximal parabolic subgroups of G G .

Injective objects in categories of monoid representations

Injective objects in categories of monoid representations

General view on monoid representations

General view on monoid representations