Mathematics

For a finite-dimensional gentle algebra, it is already known that the functorially finite torsion classes of its category of finite-dimensional modules can be classified using a combinatorial interpretation, called maximal non-crossing sets of strings, of the corresponding support τ -tilting module (or equivalently, two-term silting complexes). In the topological interpretation of gentle algebras via marked surfaces, such a set can be interpreted as a dissection (or partial triangulation), or equivalently, a lamination that does not contain a closed curve. We will refine this combinatorics, which gives us a classification of torsion classes in the category of finite length modules over a (possibly infinite-dimensional) gentle algebra. As a consequence, our result also unifies the functorially finite torsion class classification of finite-dimensional gentle algebras with certain classes of special biserial algebras - such as Brauer graph algebras.

The Clebsh-Gordan coefficients for the Lie algebra gl 3 in the Gelfand-Tsetlin base are calculated. In contrast to previous papers the result is given as an explicit formula. To obtain the result a realization of a representation in the space of functions on the group G L 3 is used. The keystone fact that allows to carry the calculation of Clebsh-Gordan coefficients is the theorem that says that functions corresponding to Gelfand-Tsetlin base vectors can be expressed through generalized hypergeometric functions.

We classify all irreducible coherent state representations of the holomorphic automorphism group of the tube domain over the dual of the Vinberg cone.

In this paper, we introduce the notion of BiHom-Lie conformal superalgebras. We develop its representation theory and define the cohomology group with coefficients in a module. Finally, we introduce conformal derivations of BiHom-Lie conformal superalgebras and study some of their properties.

We give a new combinatorial interpretation of Howe dual pairs of the form $(\g,{\rm Sp}_{2\ell})$, where $\g$ is a Lie (super)algebra of classical type. This is done by establising a symplectic analogue of the RSK algorithm associated to this pair, in a uniform way which does not depend on $\g$. We introduce an analogue of jeu de taquin sliding for spinor model of irreducible characters of a Lie superalgebra $\g$ to define P -tableau and show that the associated Q -tableau is given by a symplectic tableau due to King.

In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach algebra, and the techniques of Gelfand from the 1940s are applied in order to study the space of algebra homomorphisms to C . One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.

Let E/F be a finite and Galois extension of non-archimedean local fields. Let G be a connected reductive group defined over E and let M:= R E/F G be the reductive group over F obtained by Weil restriction of scalars. We investigate depth, and the enhanced local Langlands correspondence, in the transition from G(E) to M(F) . We obtain a depth-comparison formula for Weil-restricted groups.

Let F be a non-archimedean local field, A be a central simple F -algebra, and G be the multiplicative group of A . To construct types for supercuspidal representations of G , simple types by Sécherre--Stevens and Yu's construction are already known. In this paper, we compare these constructions. In particular, we show essentially tame supercuspidal representations of G defined by Bushnell--Henniart are nothing but tame supercuspidal representations defined by Yu.

Let S(g) be the symmetric algebra of a reductive Lie algebra g equipped with the standard Poisson structure. If C⊂S(g) is a Poisson-commutative subalgebra, then trdegC≤b(g) , where b(g)=(dimg+rkg)/2 . We present a method for constructing the Poisson-commutative subalgebra Z ⟨h,r⟩ of transcendence degree b(g) via a vector space decomposition g=h⊕r into a sum of two spherical subalgebras. There are some natural examples, where the algebra Z ⟨h,r⟩ appears to be polynomial. The most interesting case is related to the pair (b, u − ) , where b is a Borel subalgebra of g . Here we prove that Z ⟨b, u − ⟩ is maximal Poisson-commutative and is complete on every regular coadjoint orbit in g ∗ . Other series of examples are related to decompositions associated with involutions of g .

This paper focuses on a question raised by Holm and Jørgensen, who asked if the induced cotorsion pairs (Φ(X),Φ(X ) ??) and ( ??Ψ(Y),Ψ(Y)) in RepQA --the category of all A -valued representations of a quiver Q --are complete whenever (X,Y) is a complete cotorsion pair in an abelian category A satisfying some mild conditions. Recently, Odaba?ı gave an affirmative answer if the quiver Q is rooted and the cotorsion pair (X,Y) is further hereditary. In this paper, we significantly improve Odaba?ı's work by establishing the same result in the much more general framework of functor categories, and at the same time removing the hereditary assumption on the cotorsion pair.