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At least 19 recordsLinked to original sources

arXiv1998

Point processes and the infinite symmetric group. Part VI: Summary of results

We give a summary of the results from Parts I-V (math.RT/9804086, math.RT/9804087, math.RT/9804088, math.RT/9810013, math.RT/9810014). Our work originated from harmonic analysis on the infinite symmetric group. The problem of spectral decomposition for certain representations of this group leads to a family of probability measures on an infinite-dimensional simplex, which is a kind of dual object for the infinite symmetric group. To understand the nature of these measures we interpret them as stochastic point processes on the punctured real line and compute their correlation functions. The correlation functions are given by multidimensional integrals which can be expressed in terms of a multivariate hypergeometric series (the Lauricella function of type B). It turns out that after a slight modification (`lifting') of the processes the correlation functions take a common in Random Matrix Theory (RMT) determinantal form with a certain kernel. The kernel is expressed through the classical Whittaker functions. It depends on two parameters and admits a variety of degenerations. They include the well-known in RMT sine and Bessel kernels as well as some other Bessel-type kernels which, to our best knowledge, are new. The explicit knowledge of the correlation functions enables us to derive a number of conclusions about the initial probability measures. We also study the structure of our kernel; this finally leads to a constructive description of the initial measures. We believe that this work provides a new promising connection between RMT and Representation Theory.

math.RT↗

arXiv2006

Representations of field automorphism groups

This is a common introduction to math.RT/0101170, math.RT/0306333, math.RT/0506043, math.RT/0601028. Compared to these references there are new results including (i) a description of a separable closure of an extension of transcendence degree one of an algebraically closed field of positive characteristic; (ii) a "Künneth formula" for the products with curves; (iii) the semi-simplicity of the module of regular forms of top degree $Ω^n_{F/k}$.

math.RT↗

arXiv2026

$2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part II: Central extensions and exceptionals

Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. We continue our investigation from arXiv:2505.10683 [math.RT] into when the resulting skew-group algebra $R \ast G$ is a $3$-preprojective algebra of a $2$-representation infinite algebra, defined by a so-called cut. We consider the subgroups arising from $\operatorname{GL}_2(\mathbb{C}) \hookrightarrow \operatorname{SL}_3(\mathbb{C})$, called type (B), as well as the exceptional subgroups, called types (E) -- (L). For groups of type (B), we show that a $3$-preprojective cut exists on $R \ast G$ if and only if $G$ is not isomorphic to a subgroup of $\operatorname{SL}_2(\mathbb{C})$ or $\operatorname{PSL}_2(\mathbb{C})$. For groups $G$ of the remaining types (E) -- (L), every $R \ast G$ admits a $3$-preprojective cut, except for type (H) and (I). To prove our results for type (B), we explore how the notion of isoclinism interacts with the shape of McKay quivers. We compute the McKay quivers in detail, using a knitting-style heuristic. For the exceptional subgroups, we compute the McKay quivers directly, as well as cuts, and we discuss how this task can be done algorithmically. This provides many new examples of $2$-representation infinite algebras, and together with arXiv:2401.10720 [math.RT], arXiv:2505.10683 [math.RT] completes the classification of finite subgroups of $\operatorname{SL}_3(\mathbb{C})$ for which $R \ast G$ is a $3$-preprojective algebra.

math.RT↗

arXiv2006

L^2-cohomology of locally symmetric spaces, I

Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of X; they were introduced in [math.RT/0112251]. That paper also introduced the micro-support of an L-module, a combinatorial invariant that to a great extent characterizes the cohomology of the associated sheaf. The theory has been successfully applied to solve a number of problems concerning the intersection cohomology and weighted cohomology of the reductive Borel-Serre compactification [math.RT/0112251], as well as the ordinary cohomology of X [math.RT/0112250]. In this paper we extend the theory so that it covers L^2-cohomology. In particular we construct an L-module whose cohomology is the L^2-cohomology of X and we calculate its micro-support. As an application we obtain a new proof of the conjectures of Borel and Zucker.

math.RT↗

arXiv2006

Semicanonical basis generators of the cluster algebra of type $A_1^{(1)}$

We study the cluster variables and "imaginary" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these elements was obtained by P.Caldero and the author in math.RT/0604054. As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by G.Musiker and J.Propp in math.CO/0602408. The arguments in math.RT/0604054 used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton (math.RT/0410184). This note provides a quick, self-contained and completely elementary alternative proof of the same results.

math.RA↗

arXiv2024

Gaiotto Conjecture for $\mathrm{Rep}_q(\mathrm{F}(4))$

This paper is a part of the series proving the Gaiotto conjecture for basic classical quantum supergroups. The previous part arXiv:2107.02653 [math.RT] , arXiv:2306.09556 [math.RT], proved the Gaiotto conjecture for the general linear quantum supergroups $U_q(\mathfrak{gl}(N|M))$. Here we deal with the exceptional quantum supergroup $U_q(\mathfrak{f}(4))$.

math.RT↗

arXiv2025

Weak unipotence and Langlands duality

Weak unipotence of primitive ideals is a crucial property in the study of unitary representations of reductive groups. We establish a sufficient condition, referred to as mild unipotence, which guarantees weak unipotence and is more accessible in practice. We establish mild unipotence for both the $q$-unipotent ideals defined by McGovern and unipotent ideals attached to nilpotent orbit covers defined by Losev-Mason-Brown-Matvieievskyi (arXiv:2108.03453 [math.RT]). Our proof is conceptual and uses the bijection between special orbits in type $D$ and metaplectic special orbits in type $C$ found by Barbasch-Ma-Sun-Zhu (arXiv:2010.16089 [math.RT]) in an essential way.

math.RT↗

arXiv2000

SL(2) and z-measures

We give a representation-theoretic proof of the formula for correlation functions of z-measures obtained by Borodin and Olshanski in math.RT/9904010. This paper is historically preceding my paper math.RT/9907127.

math.RT↗

arXiv2004

Geometric rationality of equal-rank Satake compactifications

Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Satake compactification of a Hermitian locally symmetric space; Baily and Borel proved this is always geometrically rational. Satake compactifications for which all the real boundary components are equal-rank symmetric spaces are a natural generalization of the Baily-Borel-Satake compactification. Recent work (see math.RT/0112250, math.RT/0112251) indicates that this is the natural setting for many results about cohomology of compactifications of locally symmetric spaces. In this paper we prove any Satake compactification for which all the real boundary components are equal-rank symmetric spaces is geometrically rational aside from certain rational rank 1 or 2 exceptions; we completely analyze geometric rationality for these exceptional cases. The proof uses Casselman's criterion for geometric rationality. We also prove that a Satake compactification is geometrically rational if the representation is defined over the rational numbers.

math.RT↗

arXiv2004

Cluster algebras III: Upper bounds and double Bruhat cells

We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to the upper cluster algebra explicitly defined in terms of relevant combinatorial data.

math.RT↗

arXiv2003

Positivity and canonical bases in rank 2 cluster algebras of finite and affine types

The main motivation for the study of cluster algebras initiated in math.RT/0104151, math.RA/0208229 and math.RT/0305434 was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic groups. In this paper, we introduce and explicitly construct the canonical basis for a special family of cluster algebras of rank 2.

math.RT↗

arXiv2004

Quantum cluster algebras

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in math.RT/0104151; their study continued in math.RA/0208229, math.RT/0305434. This is a family of commutative rings designed to serve as an algebraic framework for the theory of total positivity and canonical bases in semisimple groups and their quantum analogs. In this paper we introduce and study quantum deformations of cluster algebras.

math.QA↗

arXiv2005

Finite dimensional representations of symplectic reflection algebras associated to wreath products II

This note extends some results of a previous paper (math.RT/0403250) about finite dimensional representations of the wreath product symplectic reflection algebra H(k,c,N,G) of rank N attached to a finite subgroup G of SL(2,C) (here k is a number and c a class function on the set of nontrivial elements of G). Specifically, let N'=(N_1,...,N_r) be a partition of N. Consider W=W_1\otimes >...\otimes W_r an irreducible representation of S_N'=S_{N_1}\times ...\times S_{N_r}\subset S_N s.t. W_i has rectangular Young diagram for any i and let {Y_i}, i=1,...,r be a collection of irreducible non isomorphic representation of the rank 1 algebra B=H(c_0,1,G) s.t. Ext^1_B(Y_i, Y_j)=0 for any i\neq j. Consider the module M'=W\otimes Y, where Y=Y_1^{N_1}\otimes ...\otimes Y_r^{N_r}, over the subalgebra S_N'#B^N \subset S_N#B^N= H(0,c_0,N,G) and form the induced module M over H(0,c_0,N,G). We show that M can be uniquely deformed along a linear subspace of codimension r in the space of the parameters (k,c) passing through c_0. This result implies the main result of math.RT/0403250 as a particular case, the case of the trivial partition N'=(N).

math.RT↗

arXiv2008

An exotic Springer correspondence for symplectic groups

This paper is a sequel to math.RT/0601155. Let G be a complex symplectic group. In math.RT/0601155, we constructed a certain G-variety N = N_1, which we call the (1-) exotic nilpotent cone. In this paper, we study the set of G-orbits of the variety N. It turns out that the variety N gives a variant of the Springer correspondence for the Weyl group of type C, but shares a similar flavor with that of type A case. (I.e. there appears no non-trivial local system and the correspondence is bijective.) As an application, we present one sufficient condition for the bijectivity of our exotic Deligne-Langlands correspondence [K1].

math.RT↗

arXiv2007

The small quantum group and the Springer resolution

In math.RT/0304173 the derived category of the principal block in modules over the Lusztig quantum algebra at a root of unity is related to the derived category of equivariant coherent sheaves on the Springer resolution. In the present paper we deduce a similar relation between the derived category of the principal block for the {\em small (reduced) quantum algebra} and the derived category of (non-equivariant) coherent sheaves on the Springer resolution. As an application we get a geometric description of Hochschild cohomology (in particular, the center) of the regular block for the small quantum group, and use it to give an explicit description of a certain subalgebra in the center (obtained previously by another method and under more restrictive assumptions in math.QA/0107098. We also briefly explain the relation of our result to the geometric description math.RT/0407048 of the derived category of modules over the De Concini -- Kac quantum algebra.

math.RT↗

arXiv1998

Point processes and the infinite symmetric group. Part I: The general formalism and the density function

We study a 2-parametric family of probability measures on an infinite-dimensional simplex (the Thoma simplex). These measures originate in harmonic analysis on the infinite symmetric group (S.Kerov, G.Olshanski and A.Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773-778). Our approach is to interpret them as probability distributions on a space of point configurations, i.e., as certain point stochastic processes, and to find the correlation functions of these processes. In the present paper we relate the correlation functions to the solutions of certain multidimensional moment problems. Then we calculate the first correlation function which leads to a conclusion about the support of the initial measures. In the appendix, we discuss a parallel but more elementary theory related to the well-known Poisson-Dirichlet distribution. The higher correlation functions are explicitly calculated in the subsequent paper (A.Borodin, math.RT/9804087). In the third part (A.Borodin and G.Olshanski, math.RT/9804088) we discuss some applications and relationships with the random matrix theory. The goal of our work is to understand new phenomena in noncommutative harmonic analysis which arise when the irreducible representations depend on countably many continuous parameters.

math.RT↗

arXiv1998

Point Processes and the Infinite Symmetric Group. Part II: Higher Correlation Functions

We continue the study of the correlation functions for the point stochastic processes introduced in Part I (G.Olshanski, math.RT/9804086). We find an integral representation of all the correlation functions and their explicit expression in terms of multivariate hypergeometric functions. Then we define a modification (``lifting'') of the processes which results in a substantial simplification of the structure of the correlation functions. It turns out that the ``lifted'' correlation functions are given by a determinantal formula involving a kernel. The latter has the form (A(x)B(y)-B(x)A(y))/(x-y), where A and B are certain Whittaker functions. Such a form for correlation functions is well known in the random matrix theory and mathematical physics. Finally, we get some asymptotic formulas for the correlation functions which are employed in Part III (A.Borodin and G.Olshanski, math.RT/9804088).

math.RT↗

arXiv1998

Point processes and the infinite symmetric group. Part III: Fermion point processes

In Part I (G.Olshanski, math.RT/9804086) and Part II (A.Borodin, math.RT/9804087) we developed an approach to certain probability distributions on the Thoma simplex. The latter has infinite dimension and is a kind of dual object for the infinite symmetric group. Our approach is based on studying the correlation functions of certain related point stochastic processes. In the present paper we consider the so-called tail point processes which describe the limit behavior of the Thoma parameters (coordinates on the Thoma simplex) with large numbers. The tail processes turn out to be stationary processes on the real line. Their correlation functions have determinantal form with a kernel which generalizes the well-known sine kernel arising in random matrix theory. Our second result is a law of large numbers for the Thoma parameters. We also produce Sturm-Liouville operators commuting with the Whittaker kernel introduced in Part II and with the generalized sine kernel.

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