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We study and classify representations of a torsion group $G$ over an idempotent semifield with special attention on the case over the Boolean semifield $\mathbb{B}$. In subsequent work we extend this theory to studying representations of…

There has been some work in the literature on limit theorems for the trace of commutators for compact Lie groups. We revisit this from the perspective of combinatorial representation theory.

The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study extension dimensions linked by recollements of derived module categories…

A representation $\rho$ of a compact group $\mathbb{G}$ selects eigenvalues if there is a continuous circle-valued map on $\mathbb{G}$ assigning an eigenvalue of $\rho(g)$ to every $g\in \mathbb{G}$. For every compact connected…

In this paper, we study Toeplitz operators on generalized flag manifolds of compact Lie groups using a representation-theoretic point of view. We prove several basic properties of these Toeplitz operators, including an abstract formula for…

We complete the determination of the $\ell$-block distribution of characters for quasi-simple exceptional groups of Lie type up to some minor ambiguities relating to non-uniqueness of Jordan decomposition. For this, we first determine the…

For an $n$-fold Kazhdan--Patterson cover or Savin's cover of a general linear group over a non-archimedean local field of residual characteristic $p$ with $\mathrm{gcd}(n,p)=1$, we realize the Gelfand--Graev representation as a Hecke…

Let $\Lambda$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of $\Lambda$-mod with $n\ge 2$. From the viewpoint of higher homological algebra, a question that naturally arose in [17] is when $\mathcal{M}$ induces…

The exceptional series is a finite list of points on a projective line with a simple Lie algebra attached to each point. This list of Lie algebras includes the five exceptional Lie algebras. We give a uniform trigonometric $R$-matrix for…

Let G be a quasi-split reductive group over a non-archimedean local field. We establish a local Langlands correspondence for all irreducible smooth complex G-representations in the principal series. The parametrization map is injective, and…

Let $\mathcal{A}$ be an abelian length category containing a $d$-cluster tilting subcategory $\mathcal{M}$. We prove that a subcategory of $\mathcal{M}$ is a $d$-torsion class if and only if it is closed under $d$-extensions and…

We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the…

As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of $q$ elements and fixed Jordan type. One obtains polynomials with respect to $q$…

Consider a strictly positively graded finitely generated infinite-dimensional real Lie algebra $\mathfrak{g}$. It has a well-defined Lie group $\overline{\mathbf{G}}$, which is an inverse limit of finite-dimensional nilpotent Lie groups (a…

Let G be a connected reductive group over a non-archimedean local field. We say that an irreducible depth-zero (complex) G-representation is non-singular if its cuspidal support is non-singular. We establish a Local Langlands Correspondence…

Support $\tau$-tilting pairs, functorially finite torsion classes and $2$-term silting complexes are three much studied concepts in the representation theory of finite-dimensional algebras, which moreover turn out to be connected via work…

Through the introduction of new ideals, and with the assistance of the $d$-th mode transition algebras $\mathfrak{A}_d$, for $d\in \mathbb{N}$, we show how Zhu's associative algebra $\mathsf{A}$, conventionally valued for tracking…

We obtain explicit formulas for the spinor representation $\rho$ of the real orthosymplectic supergroup $\mathrm{OSp}(2p|2q,\mathbb{R})$ by integral 'Gauss--Berezin' operators. Next, we extend $\rho$ to a complex domain and get a…

Let $Q$ be the Markov quiver, and let $W$ be an infinitely mutable potential for $Q$. We calculate some low degree refined BPS invariants for the resulting Jacobi algebra, and use them to show that the critical cohomological Hall algebra…

We propose axioms for 0-dimensional ideal approximation theory and note that extriangulated categories satisfy these axioms.