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.
We establish an extension of Viennot's geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type $C$ analogue of Schensted's theorem on longest decreasing subsequences. This pairs with our results from arXiv:2103.14997v1 [math.RT] on Type $C$ webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a $2k$-fold tensor product of the vector representation of $\mathfrak{sp}_{2n}$ equals the number of $(n+1)$-avoiding matchings of $2k$ points.
Let [Formula: see text] be a weighted Coxeter group such that the order [Formula: see text] of the product [Formula: see text] is not 3 for any [Formula: see text] and that [Formula: see text], where [Formula: see text] is the longest element in the parabolic subgroup [Formula: see text] of [Formula: see text] generated by [Formula: see text]. We prove that [Formula: see text] is bounded with [Formula: see text] an upper bound in the sense of Lusztig in Sec. 13.2 of [Hecke Algebras with Unequal Parameters, arXiv:math/0208154 v2 [math.RT] 10 Jun 2014], verifying a conjecture of Lusztig in our case (see Conjecture 13.4 in loc. cite).
An overview of the history of projective representations (= spin representations) of groups, preceded by the prehistory of studies on the theory of quaternion due to Rodrigues and Hamilton. Beginning with Schur, we cover many mathematicians until today, and also physicists Pauli and Dirac. This is a self translation of Appendix A of my book "Introduction to the theory of projective representations of groups" in Japanese, 2018, Sugakushobo, and may serve as an introduction to our paper arXiv: 1804.06063 [math.RT] which will appear in Kyoto J. Math.
This is a continuation of arXiv:0903.0398 [math.RT]. Let g be a simple Lie algebra. In this note, we provide simple formulae for the index of sl(2)-subalgebras in the classical Lie algebras and a new formula for the index of the principal sl(2). We also compute the difference, D, of the indices of principal and subregular sl(2)-subalgebras. Our formula for D involves some data related to the McKay correspondence for g. Using the index of sl(2)-subalgebras of classical Lie algebras, we also obtain three series of interesting combinatorial identities parameterised by partitions.
We give conceptual proofs of certain basic properties of the arrangement of shifted root hyperplanes associated to a root system and a Weyl group invariant real valued parameter function on the root system. The method is based on the role of this shifted root hyperplane arrangement for the harmonic analysis of affine Hecke algebras. In addition this yields a conceptual proof of the description of the central support of the Plancherel measure of an affine Hecke algebra given in math.RT.0101007v4.
This is a continuation of arXiv:0903.0398 [math.RT]. Let g be a simple Lie algebra. In this note, we provide simple formulae for the index of sl(2)-subalgebras in the classical Lie algebras and a new formula for the index of the principal sl(2). We also compute the difference, D, of the indices of principal and subregular sl(2)-subalgebras. Our formula for D involves some data related to the McKay correspondence for g. Using the index of sl(2)-subalgebras of classical Lie algebras, we also obtain three series of interesting combinatorial identities parameterised by partitions.
We give a complete classification of torsion pairs in the cluster category of Dynkin type An. Along the way we give a new combinatorial description of Ptolemy diagrams, an infinite version of which was introduced by Ng (1005.4364v1 [math.RT], 2010). This allows us to count the number of torsion pairs in the cluster category of type An. We also count torsion pairs up to Auslander–Reiten translation.
The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free opers for the Langlands dual group of G on the projective line with regular singularity at one point and irregular singularity of order two at another point. We also prove a multi-point generalization of this result, describing the spectra of commuting Hamiltonians in Gaudin models with irregular singulairity. In addition, we show that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of g has a cyclic vector in any irreducible finite-dimensional g-module. As a byproduct, we obtain the structure of a Gorenstein ring on any such module. This fact may have geometric significance related to the intersection cohomology of Schubert varieties in the affine Grassmannian.
In math.AG/0005152 a certain $t$-structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group was introduced (the so-called perverse $t$-structure corresponding to the middle perversity). In the present note we show that the same $t$-structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig and Vogan (and obtained by other means in math.RT/0010089).
Abstract Let k be an algebraically closed field of characteristic greater than 2, and let F=k((t)) and G=𝕊p2d. In this paper we propose a geometric analog of the Weil representation of the metaplectic group $\widetilde G(F)$. This is a category of certain perverse sheaves on some stack, on which $\widetilde G(F)$ acts by functors. This construction will be used by Lysenko (in [Geometric theta-lifting for the dual pair S𝕆2m, 𝕊p2n, math.RT/0701170] and subsequent publications) for the proof of the geometric Langlands functoriality for some dual reductive pairs.
Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let $$W=V {\oplus}Fe$$ with the form Q extending q with Q(e) = 1. Consider the standard embedding $$\mathrm{O}(V) \hookrightarrow \mathrm{O}(W)$$ and the two-sided action of $$\mathrm{O}(V)\times \mathrm{O}(V)$$ on $$\mathrm{O}(W)$$ . In this note we show that any $$\mathrm{O}(V) \times \mathrm{O}(V)$$ -invariant distribution on $$\mathrm{O}(W)$$ is invariant with respect to transposition. This result was earlier proven in a bit different form in van Dijk (Math Z 193:581–593, 1986) for $$F={\mathbb{R}}$$ , in Aparicio and van Dijk (Complex generalized Gelfand pairs. Tambov University, 2006) for $$F={\mathbb{C}}$$ and in Bosman and van Dijk (Geometriae Dedicata 50:261–282, 1994) for p-adic fields. Here we give a different proof. Using results from Aizenbud et al. (arXiv:0709.1273 (math.RT), submitted), we show that this result on invariant distributions implies that the pair (O(V), O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Fréchet representation (π, E) of $$\mathrm{O}(W)$$ we have $$ dim Hom_{\mathrm{O}(V)}(E,\mathbb{C}) \leq 1.$$ A stronger result for p-adic fields is obtained in Aizenbud et al. (arXiv:0709.4215 (math.RT), submitted).
We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is equivalent to the category of Hecke eigen-D-modules on the affine Grassmannian G((t))/G[[t]]. This conjecture was motivated by our proposal for a local geometric Langlands correspondence. In this paper we prove this conjecture for the corresponding I^0 equivariant categories, where I^0 is the radical of the Iwahori subgroup of G((t)). Our result may be viewed as an affine analogue of the equivalence of categories of g-modules and D-modules on the flag variety G/B, due to Beilinson-Bernstein and Brylinski-Kashiwara.
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.
Inspired by the results of [R. Adin, A. Postnikov, Y. Roichman, Combinatorial Gelfand model, preprint math.RT arXiv:0709.3962], we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup, and the factor power of the symmetric group. Furthermore we extend the Gelfand model for the semigroup algebras of the symmetric inverse semigroup to a Gelfand model for the $q$-rook monoid algebra.
First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field. Special case of the result is a finite presentation of the algebra of multisymmetric polynomials. Reducedness of the invariant commuting scheme is proved as a by-product. The algebra of multisymmetric polynomials over an arbitrary base ring is revisited.
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.
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).