Let $G$ be a finite reductive group defined over a finite field $F_q$. In the case where $G$ is a special linear group, we compute the multiplicities of irreducible characters of $G(F_{q^2})$ with the character of $G(F_{q^2})$ induced from the trivial character of $G(F_q)$. We discuss the relationship between these multiplicities with the theory of Shintani descent for finite reductive groups in general. We also give some formula concerning the decomposition of the character of $G(F_{q^r})$ induced from the trivial character of $G(F_q)$ for reductive groups $G$ with any positive integer $r$.
Let $K$ be a field and $G$ be a group of its automorphisms. If $G$ is precompact then $K$ is a generator of the category of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. There are non-semisimple smooth semilinear representations of $G$ over $K$ if $G$ is not precompact. In this note the smooth semilinear representations of the group $G$ of all permutations of an infinite set $S$ are studied. Let $k$ be a field and $k(S)$ be the field freely generated over $k$ by the set $S$ (endowed with the natural $G$-action). One of principal results describes the Gabriel spectr…
In this paper we give a new definition of symmetric special multiserial algebras in terms of defining cycles. As a consequence, we show that every special multiserial algebra is a quotient of a symmetric special multiserial algebra.
We provide a complete classification of all tame and wild symmetric special multiserial algebras in terms of the underlying Brauer configuration. Our classification contains the symmetric special multiserial algebras of finite representation type.
Let $\mathbf{k}$ be an algebraically closed field. In this article, inspired by the description of indecomposable objects in the derived category of a gentle algebra obtained by V. Bekkert and H. A. Merklen, we define string complexes for a certain class $\mathscr{C}$ of symmetric special biserial algebras, which are indecomposable perfect complexes in the corresponding derived category. We also prove that if $Λ$ is a $\mathbf{k}$-algebra in the class $\mathscr{C}$ and $P^\bullet$ is a string complex over $Λ$, then $P^\bullet$ lies in the rim of its Auslander-Reiten component.
Let $\k$ be an algebraically closed field, let $\A$ be a finite dimensional $\k$-algebra and let $V$ be a $\A$-module with stable endomorphism ring isomorphic to $\k$. If $\A$ is self-injective then $V$ has a universal deformation ring $R(\A,V)$, which is a complete local commutative Noetherian $\k$-algebra with residue field $\k$. Moreover, if $Λ$ is also a Frobenius $\k$-algebra then $R(\A,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $\Ar$-modules whose stable endomorphism ring isomorphic to $\k$, where $\Ar$ is a symmetric special b…
By a result claimed by Pogorzały selfinjective special biserial algebras can be stably equivalent only to stably biserial algebras and these two classes coincide. By an example of Ariki, Iijima and Park the classes of stably biserial and selfinjective special biserial algebras do not coincide. In these notes we provide a detailed proof of the fact that a selfinjective special biserial algebra can be stably equivalent only to a stably biserial algebra following some ideas from the paper by Pogorzały. We will analyse the structure of symmetric stably biserial algebras and show that in characteri…
We introduce Brauer complex of symmetric SB-algebra, and reformulate in terms of Brauer complex the so far known invariants of stable and derived equivalence of symmetric SB-algebras. In particular, the genus of Brauer complex turns out to be invariant under derived equivalence. We study transformations of Brauer complexes which preserve class of derived equivalence. Additionally, we establish a new invariant of derived equivalence of symmetric SB-algebras. As a consequence, symmetric SB-algebras with Brauer complex of genus 0 are classified. Keywords: Brauer tree algebras, special biserial al…
The notion of mutation plays crucial roles in representation theory of algebras. Two kinds of mutation are well-known: tilting/silting mutation and quiver-mutation. In this paper, we focus on tilting mutation for symmetric algebras. Introducing mutation of SB quivers, we explicitly give a combinatorial description of tilting mutation of symmetric special biserial algebras. As an application, we generalize Rickard's star theorem. We also introduce flip of Brauer graphs and apply our results to Brauer graph algebras.
We give a formula for the character of the representation of the symmetric group $S_n$ on each isotypic component of the cohomology of the set of regular elements of a maximal torus of $SL_n$, with respect to the action of the centre.
We consider a natural generalisation of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the algebras of [Bocian-Holm-Skowroński, J. Pure Appl. Algebra 2004], where they study the weakly symmetric algebras of Euclidean type, as well as some algebras of dihedral type.
We provide a combinatorial algorithm for constructing the stable Auslander-Reiten component containing a given indecomposable module of a symmetric special biserial algebra using only information from its underlying Brauer graph. We also show that the structure of the Auslander-Reiten quiver is closely related to the distinct Green walks around the Brauer graph and detail the relationship between the precise shape of the stable Auslander-Reiten components for domestic Brauer graph algebras and their underlying graph. Furthermore, we show that the specific component containing a given simple or…
We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and Kasten-Schmidt. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands functoriality one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blas…
We determine when an antiinvolution on an adjoint semisimple linear algebraic group extends to an antiinvolution on a $J$-irreducible monoid. Using this information, we study a special class of compactifications of symmetric varieties. Extending the work of Springer on involutions, we describe the parametrizing sets of Borel orbits in these special embeddings.
We formulate a theory of invariants for the spin symmetric group in some suitable modules which involve the polynomial and exterior algebras. We solve the corresponding graded multiplicity problem in terms of specializations of the Schur Q-functions and a shifted q-hook formula. In addition, we provide a bijective proof for a formula of the principal specialization of the Schur Q-functions.
We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra $gl_N$. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.
We define a special sort of weighted oriented graphs, signed quivers. Each of these yields a symmetric quiver, i.e., a quiver endowed with an involutive anti-automorphism and the inherited signs. We develop a representation theory of symmetric quivers, in particular we describe the indecomposable symmetric representations. Their dimensions constitute root systems corresponding to certain symmetrizable generalized Cartan matrices.
We construct special solutions of the quantum Knizhnik-Zamolodchikov equation on the tensor product of the vector representation of the quantum algebra of type A_{N-1}. They are constructed from non-symmetric Macdonald polynomials through the action of the affine Hecke algebra. As a special case the matrix element of the vertex operators of level one is reproduced.
