Two contributions to the representation theory of algebraic groups

manuscripta mathematica, 2020

In this paper, using computations done through the LiE software, we compare tensor product of irreducible selfdual representations of the special linear group with those of classical groups to formulate some conjectures relating the two. In the process a few other phenomenon present themselves which we record as questions. More precisely, under the natural correspondence of irreducible finite dimensional selfdual representations of SL 2n (C) with those of Spin 2n+1 (C), it is easy to see that if the tensor product of three irreducible representations of Spin 2n+1 (C) contains the trivial representation, then so does the tensor product of the corresponding representations of SL 2n (C). The paper formulates a conjecture in the reverse direction. We also deal with the pair (SL 2n+1 (C), Sp 2n (C)). March 24, 2020 CONTENTS 1. Introduction 1 2. Relating multiplicities 2 3. Review of the group G σ 4 4. An example of twisted character 5 5. The pair (SL 2n (C), Spin 2n+1 (C)) 6 6. The pair (SL 2n+1 (C), Sp 2n (C)) 12 7. Some questions relating B n , C n and A 2n , A 2n+1 14 8. Sample computations 15 References 19 ⊂ π Spin 1 ⊗ π Spin 2 , of Spin 2n+1 (C), appearing with multiplicity m(π Spin 3 , π Spin 1 ⊗ π Spin 2 ) = 0, we have: However, what came as quite a bit of surprise, after much computer assisted checks, that the converse of the above assertion, i.e., 2 ) = 0, holds, or rather, almost holds, which egged our curiosity on, and in the process we hope we have found something of some interest. We end the introduction by mentioning that irreducible components of the tensor product of two irreducible representations of a simple group are reasonably well understood through the 'Saturation conjecture', a theorem for SL n (C) due to Knutson and Tao [KT], and due to several works of P. Belkale, J. Hong, S. Kumar and L. Shen, among others, for classical groups, see e.g. [BK], [HS], [H], [Ku2], [Ku-St], and [Ku] for a general survey. However, the more precise question on what the multiplicities look like has not been attempted as much. Looking at tables of tensor product multiplicities, one is dazzled by the wealth of data on statistical behaviour of multiplicities it contains. No precise question has been attempted as far as we know. In our work relating tensor product of SL 2n (C) and Spin 2n+1 (C) (and more generally a pair of groups G, G σ that we will soon come to), we are concerned (though not this very precise question) with how often are multiplicities in the tensor product for SL 2n (C) are even versus odd: a well-known question in the theory of partition functions which is very much present in representation theory of SL 2n (C) through Kostant partition function. Let G be a connected reductive algebraic group over C. Assume that σ is a diagram automorphism of G of order 2. Thus we assume that (B, T, S) is a fixed Borel subgroup B in G, containing a maximal torus T , and with S a fixed pinning on B, which is an identification of each simple root of T on the Lie algebra of B with the additive group C, and that (B, T, S) is left invariant under σ. Suppose that π is an irreducible representation of G which is invariant under σ, and thus extends to a natural representation of G = G ⋊ σ by demanding that the action of σ on the highest B-weight of π is trivial. Denote the representation of G so obtained as π. The representation π of G gives rise to an irreducible representation of the group G σ

2018

We study the quotient of $\mathcal{T}_n = Rep(GL(n|n))$ by the tensor ideal of negligible morphisms. If we consider the full subcategory $\mathcal{T}_n^+$ of $\mathcal{T}_n$ of indecomposable summands in iterated tensor products of irreducible representations up to parity shifts, its quotient is a semisimple tannakian category $Rep(H_n)$ where $H_n$ is a pro-reductive algebraic group. We determine the connected derived subgroup $G_n \subset H_n$ and the groups $G_{\lambda} = (H_{\lambda})_{der}^0$ corresponding to the tannakian subcategory in $Rep(H_n)$ generated by an irreducible representation $L(\lambda)$. This gives structural information about the tensor category $Rep(GL(n|n))$, including the decomposition law of a tensor product of irreducible representations up to summands of superdimension zero. Some results are conditional on a hypothesis on $2$-torsion in $\pi_0(H_n)$.

We study the structure of the indecomposable direct summands of tensor products of two restricted rational simple modules for the algebraic group SL3(K), where K is an algebraically closed field of characteristic p ≥ 5. We also give a characteristicfree algorithm for the decomposition of such a tensor product into indecomposable direct summands. The p < 5 case was studied in the authors' earlier paper . We find that for characteristics p ≥ 5 all the indecomposable summands are rigid, in contrast to the characteristic 3 case.

Comptes Rendus Mathematique, 2004

We show that the vector space of fixed valence Killing tensors on a space of constant curvature is naturally isomorphic to a certain highest weight, irreducible representation of the general linear group. The isomorphism is equivariant in the sense that the natural action of the isometry group corresponds to the restriction of the linear action to the appropriate subgroup. As an application, we deduce the Delong-Takeuchi-Thompson formula on the dimension of the vector space of Killing tensors from the classical Weyl dimension formula. To cite this article: R.G.

The aim of the current dissertation is to address certain problems in the representation theory of simple Lie algebras and associated quantum algebras. In Part I, we study the simple Lie group of type G 2 from the cluster point of view. We prove a conjecture of Geiss, Leclerc and Schröer [25], relating the geometry of the partial flag varieties to cluster algebras in the case of G 2. In Part II, we establish a concrete relationship between certain infinite dimensional quantum groups, namely the quantum loop algebra U (Lg) and the Yangian Y (g), associated with a simple Lie algebra g. The main result of this part gives a construction of an explicit isomorphism between completions of these algebras, thus strengthening the well known Drinfeld's degeneration homomorphism [10, 27]. In Part III, we give a proof of the monodromy conjecture of Toledano Laredo [50] for the case of g = sl 2. The monodromy conjecture relates two classes of representations of the affine braid group, the first arising from the quantum Weyl group operators of the quantum loop algebra and the second coming from the monodromy of the trigonometric Casimir connection, a flat connection introduced by Toledano Laredo [50]. To Prof. Valerio Toledano Laredo, for teaching me the monodromy side of quantum groups, for your constant availability and for having great patience with me during all our joint projects.

Springer eBooks, 1980

Il Nuovo Cimento B, 1981

ABSTRACT As a development of a preceding paper we obtain the representations of projective Lie algebras by contraction of representations of vector algebras of the same dimension. As examples, we deduce the projective representations of the Galilei unidimensional group and ofIO 2. In addition, we discuss the possibility that the use of the contraction allows us to classify completely the Lie groups and their representations. Come evoluzione della tecnica sviluppata in un precedente articolo, si ottengono le rappresontazioni delle algebre di Lie proiettive mediante contrazione di rappresentazioni di algebre vettoriali della stessa dimensione. La procedura è illustrata con le rappresentazioni proiettive del gruppo di Galileo unidimensionale e diIO 2. Si discute, inoltre, la possibilitá che la contrazione permetta di affrontare il problema della classificazione dei gruppi di Lie e delle loro rappresentazioni. B развитие предыдущей статьи мы получаем представления проек тивных алгебр Ли в реузльтат е сокращения предста влений векторных алгебр тог о же числа измерений. К ак пример, мы выводим п роективные представ ления одномерно мы выводим проективн ые представления одн омерной группы Галилея и /02. Кро ме того, мы обсуждаем возможность того, что использование сокра щения позволяет нам п олностью классифицировать гр уппы Ли и их представл ения.

Journal of Algebra, 2005

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E 9 ). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann's path algorithm and conforms with the uniform combinatorial behavior recently discovered by H. Wenzl for the series EN , N = 9. 2

Journal of the American Mathematical Society, 1994

Arkiv för matematik, 1972