We consider the matrix spherical function related to the compact symmetric pair $(G,K)=(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$. The irreducible $K$ representations $(π,V)$ in the ${\rm U}(n)$ part are considered and the induced representation $\mathrm{Ind}_K^Gπ$ splits multiplicity free. In this case, the irreducible $K$ representations in the ${\rm U}(n)$ part are studied. The corresponding spherical functions can be approximated in terms of the simpler matrix-valued functions. We can determine the explicit spherical functions using the action of a differential operator. We consider several cases of irreducible $K$ representations and the orthogonality relations are also described.
In this article, using the notion of group contraction, we obtain the spherical functions of the strong Gelfand pair $(\mathrm{M}(n),\mathrm{SO}(n))$ as an appropriate limit of spherical functions of the strong Gelfand pair $(\mathrm{SO}(n+1),\mathrm{SO}(n))$ and also of the strong Gelfand pair $(\mathrm{SO}_0(n,1),\mathrm{SO}(n))$.
For a semisimple complex Lie algebra $\mathfrak g$, the BGG category $\mathcal{O}$ is of particular interest in representation theory. It is known that Irving's shuffling functors $\mathrm{Sh}_{w}$, indexed by elements $w\in W$ of the Weyl group, induce an action of the braid group $B_W$ associated to $W$ on the derived categories $D^\mathrm{b}(\mathcal{O}_λ)$ of blocks of $\mathcal{O}$. We show that for maximal parabolic subalgebras $\mathfrak{p}$ of $\mathfrak{sl}_n$ corresponding to the parabolic subgroup $W_\mathfrak{p}=S_{n-1}\times S_1$ of $S_n$, the derived shuffling functors $\mathbf{L…
We propose a general algorithm to compute all the symmetry classes of any $\mathrm{SO}(3)$ or $\mathrm{O}(3)$ linear representation. This method relies on the introduction of a binary operator between sets of conjugacy classes of closed subgroups, called the clips. We compute explicit tables for this operation which allows to solve definitively the problem.
We compute the total Stiefel Whitney class for a real representation $π$ of $\mathrm{GL}_n(\mathbb{F}_q)$, where $q$ is odd in terms of character values of $π$ on order $2$ diagonal elements. We also compute the total Stiefel Whitney class of real representations of $\mathrm{GL}_n(\mathbb{R})$ and $\mathrm{GL}_n(\mathbb{C})$.
We compute the total Stiefel Whitney class for a real representation $π$ of $\mathrm{GL}_2(\mathbb{F}_q)$, where $q$ is odd. The obstruction class of $π$ is defined to be the Stiefel Whitney class of lowest positive degree that does not vanish. We provide an expression for the obstruction class of $π$ in terms of its character values if $\detπ=1$.
In the present paper, we construct two classes of non-weight modules $Ω(λ,α,β)\otimes\mathrm{Ind}(M)$ and $\mathcal{M}\big(V,Ω(λ,α,β)\big)$ over the twisted Heisenberg-Virasoro algebra, which are both associated with the modules $Ω(λ,α,β)$. We present the necessary and sufficient conditions under which modules in these two classes are irreducible and isomorphic, and also show that the irreducible modules in these two classes are new. Finally, we construct non-weight modules $\mathrm{Ind}_{\underline y,λ}(\C_{RS})$ and $\mathrm{Ind}_{\underline z,λ}(\C_{PQ})$ over the twisted Heisenberg-Virasor…
We give a combinatorial description of all affine spherical varieties with prescribed weight monoid $Γ$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_Γ$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_Γ$ to be irreducible and exhibit several examples where $\mathrm M_Γ$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_Γ$.
The non-spherical sup-norm problem for $\mathrm{GL}(n)$ and generalized spherical functions
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；MTA–HUN-REN RI Lendület Automorphic Research Group（匈牙利科学院-匈牙利研究与创新网络雷内利特自守群研究组）；MTA–HUN-REN RI Lendület Analytic Number Theory and Representation Theory Research Group（匈牙利科学院-匈牙利研究与创新网络雷内利特解析数论与表示论研究组）；Bryn Mawr College（布林茅尔学院）
Let $Λ$ be an Artin algebra. In 2014, T. Adachi, O. Iyama and I. Reiten proved that the torsion funtorially finite classes in $\mathrm{mod}\,(Λ)$ can be described by the $τ$-tilting theory. The aim of this paper is to introduce the notion of $F$-torsion class in $\mathrm{mod}\,(Λ)$, where $F$ is an additive subfunctor of $\mathrm{Ext}^1_Λ,$ and to characterize when these clases are preenveloping and $F$-preenveloping. In order to do that, we introduce the notion of $F$-presilting $Λ$-module. The latter is both a generalization of $τ$-rigid and $F$-tilting in $\mathrm{mod}(Λ).$
Given a connected reductive algebraic group $G$ and a finitely generated monoid $Γ$ of dominant weights of $G$, in 2005 Alexeev and Brion constructed a moduli scheme $\mathrm M_Γ$ for multiplicity-free affine $G$-varieties with weight monoid $Γ$. This scheme is equipped with an action of an `adjoint torus' $T_{\mathrm{ad}}$ and has a distinguished $T_{\mathrm{ad}}$-fixed point $X_0$. In this paper, we obtain a complete description of the $T_{\mathrm{ad}}$-module structure in the tangent space of $\mathrm M_Γ$ at $X_0$ for the case where $Γ$ is saturated. Using this description, we prove that t…
We show that, for a sheet or a Lusztig stratum S containing spherical conjugacy classes in a connected reductive algebraic group G over an algebraically closed field in good characteristic, the orbit space S/G is isomorphic to the quotient of an affine subvariety of G modulo the action of a finite abelian 2-group. The affine subvariety is a closed subset of a Bruhat double coset and the abelian group is a finite subgroup of a maximal torus of G. We show that sheets of spherical conjugacy classes in a simple group are always smooth and we list which strata containing spherical classes are smoot…
We prove that for a simply laced group, the closure of the Borel conjugacy class of any nilpotent element of height $2$ in its conjugacy class is normal and admits a rational resolution. We extend this, using Frobenius splitting techniques, to the closure in the whole Lie algebra if either the group has type $A$ or the element has rank $2$.
In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of $\mathrm{SL}_2(\mathbb{C})$-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic punctures are chosen to be semi-simple non-generic, we show that a new interaction phenomenon appears generating a recursive pattern.
In the present paper we further the study of the compression cone of a real spherical homogeneous space $Z=G/H$. In particular we provide a geometric construction of the little Weyl group of $Z$ introduced recently by Knop and Krötz. Our technique is based on a fine analysis of limits of conjugates of the subalgebra $\mathrm{Lie}(H)$ along one-parameter subgroups in the Grassmannian of subspaces of $\mathrm{Lie}(G)$. The little Weyl group is obtained as a finite reflection group generated by the reflections in the walls of the compression cone.
In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are u…
We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair $(\mathrm{U}(n) \ltimes \mathrm{Herm}(n), \mathrm{U}(n))$. It is inspired by a similar approach of Kieburg and Kösters for products of random matrices. The spherical functions have determinantal expressions because of the Harish-Chandra/Itzykson-Zuber integral formula. It leads to remarkably simple expressions for the spherical transform and its inverse. The spherical transform is applied to sums of unitarily invariant random matrices from polynomial ensembles and the subclass…
We prove that for any automorphism $ϕ$ of the restricted wreath product $\mathbb{Z}_2 \mathrm{wr} \mathbb{Z}^k$ and $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d}$ the Reidemeister number $R(ϕ)$ is infinite, i.e. these groups have the property $R_\infty$. For $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d+1}$ and $\mathbb{Z}_p \mathrm{wr} \mathbb{Z}^k$, where $p>3$ is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property $R_\infty$. For these groups and $\mathbb{Z}_m \mathrm{wr} \mathbb{Z}$, where $m$ is relatively prime to $6$, we prove the t…
The notion of a $(k,n)$-frieze pattern was introduced by the author as a generalisation of the classical frieze patterns. In this article we describe connections between classes of $(3,n)$-frieze patterns and classes of $\mathrm{SL}_3$-frieze patterns. We introduce the idea of a superimposed triangulation and clarify how superimposed triangulations may be used to understand quiddity sequences for $\mathrm{SL}_3$-frieze patterns.
We prove several reality properties for finite simple orthogonal groups. For any prime power $q$ and $m\geq 1$, we show that all real conjugacy classes are strongly real in the simple groups $\mathrm{P}Ω^{\pm}(4m+2,q), m \geq 1$, except in the case $\mathrm{P}Ω^{-}(4m+2,q)$ with $q \equiv 3(\mathrm{mod} \; 4)$, and we construct weakly real classes in this exceptional case for any $m$. We also show that no irreducible complex character of $\mathrm{P}Ω^{\pm}(n,q)$ can have Frobenius-Schur indicator $-1$, except possibly in the case $\mathrm{P}Ω^{-}(4m+2,q)$ with $q \equiv 3(\mathrm{mod} \; 4)$.
Let $Λ_Q$ be the preprojective algebra of a finite acyclic quiver $Q$ of non-Dynkin type and $D^b(\mathrm{rep}^n Λ_Q)$ be the bounded derived category of finite dimensional nilpotent $Λ_Q$-modules. We define spherical twist functors over the root category $\mathcal{R}_{Λ_Q}$ of $D^b(\mathrm{rep}^n Λ_Q)$ and then realize the Weyl group associated to $Q$ as certain subquotient of the automorphism group of the Ringel-Hall Lie algebra $\mathfrak{g}(\mathcal{R}_{Λ_Q})$ of $\mathcal{R}_{Λ_Q}$ induced by spherical twist functors. We also present a conjectural relation between certain Lie subalgebras …
We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $f \colon X \to \mathrm{Spec}\ R$, where $X$ has only Gorenstein terminal singularities, there is an associated finite dimensional algebra $A_{\mathrm{con}}$ known as the contraction algebra. As a corollary of our main result, th…
We consider orthogonally invariant probability measures on $\mathrm{GL}_n(\mathbb{R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices to the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu-Shub\cite{DS}. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.
