Let $G$ be a simple algebraic group in defining characteristic $p>0$, and let $V$ be an irreducible $G$-module which is the tensor product of exactly two non-trivial modules. We obtain a criterion for $V$ to have the zero weight. In addition, we provide a uniform criterion for an irreducible representation of a simple Lie algebra over the complex numbers to have a multiple of a prescribed fundamental weight.
We give a complete classification of reductive symmetric pairs (g, h) with the following property: there exists at least one infinite-dimensional irreducible (g,K)-module X that is discretely decomposable as an (h,H \cap K)-module. We investigate further if such X can be taken to be a minimal representation, a Zuckerman derived functor module A_q(λ), or some other unitarizable (g,K)-module. The tensor product $π_1 \otimes π_2$ of two infinite-dimensional irreducible (g,K)-modules arises as a very special case of our setting. In this case, we prove that $π_1 \otimes π_2$ is discretely decomposa…
We present (with proof) a new family of decomposable Specht modules for the symmetric group in characteristic 2. These Specht modules are labelled by partitions of the form $(a,3,1^b)$, and are the first new examples found for thirty years. Our method of proof is to exhibit summands isomorphic to irreducible Specht modules, by constructing explicit homomorphisms between Specht modules.
In this note, we determine the irreducible characters for the simple algebraic groups of type $A_5$ over an algebraically closed field $K$ of characteristic 3, by using a theorem of Xi Nanhua and the Matlab software. In order to obtain higher speed than in Jiachen et al we modify the algorithm to compute the irreducible characters.
Let $G$ be an algebraic group of classical type of rank $l$ over an algebraically closed field $K$ of characteristic $p$. We list and determine the dimensions of all irreducible $KG$-modules $L$ with $\dim L < \binom{l+1}{4}$ if $G$ is of type $A_l$, and with $\dim L < 16 \binom{l}{4}$, if $G$ is of type $B_l$, $C_l$ or $D_l$.
We give a cohomological treatment of a character theory for (g,K)-modules. This leads to a nice formalism extending to large categories of not necessarily admissible (g,K)-modules. Due to results of Hecht, Schmid and Vogan the classical results of Harish-Chandra's global character theory extend to this general setting. As an application we consider a general setup, for which we show that algebraic characters answer discretely decomposable branching problems.
Let $n\geq 3,$ and let $Y$ be a simply connected, simple algebraic group of type $D_{n+1}$ over an algebraically closed field $K.$ Also let $X$ be the subgroup of type $B_n$ of $Y,$ embedded in the usual way. In this paper, we correct an error in a proof of a theorem of Seitz, resulting in the discovery of a new family of triples $(X,Y,V),$ where $V$ denotes a finite-dimensional, irreducible, rational $KY$-module, on which $X$ acts irreducibly. We go on to investigate the impact of the existence of the new examples on the classification of the maximal closed connected subgroups of the classica…
Assume that $k$ is an algebraically closed field and $A$ is a finite-dimensional wild $k$-algebra. Recently, L. Gregory and M. Prest proved that in this case the width of the lattice of all pointed $A$-modules is undefined and hence there exists a super-decomposable pure-injective $A$-module, if the base field $k$ is countable. Here we give a proof of a stronger result. Namely, we show that there exists a special family of pointed $A$-modules, called an independent pair of dense chains of pointed modules. This also yields that the width of the lattice of all pointed $A$-modules is undefined.
Let G be an almost simple, simply connected algebraic group over an algebraically closed field of characteristic p>0. In this paper we restate our conjecture from 1979 on the characters of irreducible modular representations of G so that it is now directly applicable to any dominant highest weight.
Let G be a simple, simply connected and connected algebraic group over an algebraically closed field of characteristic p>0, and let V be a rational G-module such that dim V <= p. According to a result of Jantzen, V is completely reducible, and H^1(G,V)=0. In this paper we show that H^2(G,V) = 0 unless some composition factor of V is a non-trivial Frobenius twist of the adjoint representation of G.
The behavior of the images of a fixed element of order p in irreducible representations of a classical algebraic group in odd characteristic p with highest weights large enough with respect to p and this element is investigated. Lower estimates for the number of Jordan blocks of size p in images of such elements that lie in naturally embedded subgroups of the same type as the initial group and smaller ranks are obtained.
Let $k$ be an algebraically closed field of characteristic $p \geq 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\ell$ and let $T$ be a maximal torus in $G$ with rational character group $X(T)$. For a nonzero $p$-restricted dominant weight $λ\in X(T)$, let $V$ be the associated irreducible $kG$-module. Define $ν_{G}(V)$ to be the minimum codimension of eigenspaces corresponding to non-central elements of $G$ on $V$. In this paper, we calculate $ν_{G}(V)$ for $G$ of type $A_{\ell}$, $\ell \geq 16$, and $dim(V) \leq \frac{\ell^{3}}{2}$; for $G$ of type $B_{\e…
In this paper we work in the `split' discrete Clifford analysis setting, i.e. the m-dimensional function theory concerning null-functions, defined on the grid Z^m, of the discrete Dirac operator D, involving both forward and backward differences, which factorizes the (discrete) Star-Laplacian (Delta = D^2). We show how the space M_k of discrete homogeneous spherical monogenics of degree k, is decomposable into 2^{2m-n} isomorphic irreducible representations with highest weight (k + 1/2, 1/2,...,1/2) in the odd-dimensional case and two times 2^{2m-n} isomorphic irreducible representations with …
Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module. We say that $(G,H,V)$ is an irreducible triple if $V$ is irreducible as a $KH$-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups…
Let $H$ be a linear algebraic group whose connected component $G\neq 1$ is simple and $H/G$ is cyclic. We determine the irreducible projective representations $ϕ$ of $H$ such that $ϕ(G)$ is irreducible and $ϕ(h)$ has simple spectrum for some $h\in H$. The latter means that all irreducible constituents of the group $ϕ( \langle h\rangle)$ are of multiplicity 1. (Here $\langle h\rangle$ is the subgroup of $H$ generated by $h$.) This extends an earlier known result for $H=G$.
We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers.
Let $G$ be a simple classical algebraic group over an algebraically closed field $K$ of characteristic $p \ge 0$ with natural module $W$. Let $H$ be a closed subgroup of $G$ and let $V$ be a non-trivial irreducible tensor-indecomposable $p$-restricted rational $KG$-module such that the restriction of $V$ to $H$ is irreducible. In this paper we classify all such triples $(G,H,V)$, where $H$ is a maximal closed disconnected positive-dimensional subgroup of $G$, and $H$ preserves a natural geometric structure on $W$.
Let G be a simple classical algebraic group over an algebraically closed field K of characteristic $p \ge 0$ with natural module W. Let H be a closed subgroup of G and let V be a nontrivial p-restricted irreducible tensor indecomposable rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V) of this form, where $V \ne W, W^{*}$ and H is a disconnected almost simple positive-dimensional closed subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classification of the irreducible …
For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the alternating or symmetric square representation is irreducible or decomposes into an irreducible representation and a trivial representation.
Let $K$ be an algebraically closed field of characteristic $p\geqslant 0$ and let $Y=\mbox{Spin}_{2n+1}(K)$ $(n\geqslant 3)$ be a simply connected simple algebraic group of type $B_n$ over $K.$ Also let $X$ be the subgroup of type $D_n,$ embedded in $Y$ in the usual way, as the derived subgroup of the stabilizer of a non-singular one-dimensional subspace of the natural module for $Y.$ In this paper, we give a complete set of isomorphism classes of finite-dimensional, irreducible, rational $KY$-modules on which $X$ acts with exactly two composition factors.
The classical McKay correspondence for finite subgroups $G$ of $\SL(2,\C)$ gives a bijection between isomorphism classes of nontrivial irreducible representations of $G$ and irreducible components of the exceptional divisor in the minimal resolution of the quotient singularity $\A^2_\C/G$. Over non algebraically closed fields $K$ there may exist representations irreducible over $K$ which split over $\bar{K}$. The same is true for irreducible components of the exceptional divisor. In this paper we show that these two phenomena are related and that there is a bijection between nontrivial irreduc…
We study the spectra of non-regular semisimple elements in irreducible representations of simple algebraic groups. More precisely, we prove that if G is a simply connected simple linear algebraic group and f is a non-trivial irreducible representation of G in some GL(V) for which there exists a non-regular non-central semisimple element s in G such that f(s) has almost simple spectrum, then, with few exceptions, G is of classical type and dim V is minimal possible. Here the spectrum of a diagonalizable matrix is called simple if all eigenvalues are of multiplicity 1, and almost simple if at mo…
Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p\geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial irreducible $KG$-module, which is $p$-restricted, tensor indecomposable and rational. Assume that the restriction of $V$ to $H$ is irreducible. In this paper, we study the triples $(G,H,V)$ of this form when $G$ is a classical group and $H$ is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples $(G,H,V)$ to the case w…
Let $G$ be a finite group, $H$ be a normal subgroup of prime index $p$. Let $F$ be a field of either characteristic $0$ or prime to $|G|$. Let $η$ be an irreducible $F$-representation of $H$. If $F$ is an algebraically closed field of characteristic either $0$ or prime to $|G|$, then the induced representation $η\uparrow^{G}_{H}$ is either irreducible or a direct sum of $p$ pairwise inequivalent irreducible representations. In this paper, we show that if $F$ is not assumed algebraically closed field, then there are five possibilities in the decomposition of induced representation into irreduci…
In this paper we show that the irreducible representations of a finite inverse semigroup $S$ over an algebraically closed field $F$ are in bijection with the conjugacy classes of $S$ if the characteristic of $F$ is zero or a prime number that does not divide the order of any maximal subgroup of $S$.
Let $G$ be a simple linear algebraic group defined over an algebraically closed field of characteristic $p\geq 0$ and let $ϕ$ be a $p$-restricted irreducible representation of $G$. Let $T$ be a maximal torus of $G$ and $s\in T$. We say that $s$ is strongly regular if $α(s)\neβ(s)$ for all distinct $T$-roots $α$ and $β$ of $G$. Our main result states that if all but one of the eigenvalues of $ϕ(s)$ are of multiplicity 1 then, with a few specified exceptions, $s$ is strongly regular. This can be viewed as an extension of our earlier result saying that under the same hypotheses, $s$ must be regul…
Let $G$ be a split reductive group over a local field $\bK$, and let $G((t))$ be the corresponding loop group. In \cite{GK} we have introduced the notion of a representation of (the group of $\bK$-points) of $G((t))$ on a pro-vector space. In addition, we have defined an induction procedure, which produced $G((t))$-representations from usual smooth representations of $G$. We have conjectured that the induction of a cuspidal irreducible representation of $G$ is irreducible. In this paper we prove this conjecture for $G=SL_2$.
This work completes the classification of the imprimitive irreducible modules, over algebraically closed fields of characteristic 0, of the finite quasisimple groups.
In the representation theory of split reductive algebraic groups, it is well known that every Weyl module with minuscule highest weight is irreducible over every field. Also, the adjoint representation of $E_8$ is also irreducible over every field. In this paper, we prove a converse to these statements, as conjectured by Gross: if a Weyl module is irreducible over every field, it must be either one of these, or trivially constructed from one of these.
We give a complete description of the irreducible representations of algebraic fusions of association schemes, in terms of the irreducible representations of a Schur cover of the corresponding group of algebraic automorphisms.
We discuss recent developments on branching problems of irreducible unitary representations $π$ of real reductive groups when restricted to reductive subgroups. Highlighting the case where the underlying $(g,K)$-modules of $π$ are isomorphic to Zuckerman's derived functor modules $A_q(λ)$, we show various and rich features of branching laws such as infinite multiplicities, irreducible restrictions, multiplicity-free restrictions, and discrete decomposable restrictions. We also formulate a number of conjectures.
We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change, tensor products of irreducible and Weyl modules, and the block decomposition of the underlying abelian category. Several results are interestingly related to the study of irreducible representations of polynomial algebras and Galois theory.
We construct some irreducible representations of the Leavitt path algebra of an arbitrary quiver. The constructed representations are associated to certain algebraic branching systems. For a row-finite quiver, we classify algebraic branching systems, to which irreducible representations of the Leavitt path algebra are associated. For a certain quiver, we obtain a faithful completely reducible representation of the Leavitt path algebra. The twisted representations of the constructed ones under the scaling action are studied.
In this paper, we classify the finite dimensional irreducible modules for affine BMW algebra over an algebraically closed field with arbitrary characteristic.
Let L be a finite-dimensional Lie algebra over a field of non-zero characteristic. By a theorem of Jacobson, L has a finite-dimensional faithful module which is completely reducible. We show that if the field is not algebraically closed, then L has an irreducible such module. We also give a necessary and sufficient condition for a finite-dimensional Lie algebra over an algebraically closed field of non-zero characteristic to have a faithful irreducible module.
In this article a complete description is given of the simple representations of a 3-dimensional Sklyanin algebra associated to a torsion point. In order to determine these irreducible representations, a review is given of classical results regarding representation theory of graded rings with excellent homological and algebraic properties.
We use tilting modules to study the structure of the tensor product of two simple modules for the algebraic group $\SL_2$, in positive characteristic, obtaining a twisted tensor product theorem for its indecomposable direct summands. Various other related results are obtained, and numerous examples are computed.
We construct a moduli space for the connected subgroups of an algebraic group and the corresponding universal family. Morphisms from an algebraic variety to this moduli space correspond to flat families of connected algebraic subgroups parametrised by this variety. This moduli space is obtained by gluing together infinitely many irreducible projective varieties of bounded dimension along closed subvarieties. Regarding families of non-connected subgroups of an algebraic group, we show that, given sich a family, the corresponding family of identity components is an irreducible component of the f…
In this paper we show how a theorem of Jantzen relating the character of a finite dimensional irreducible representation of a disconnected semisimple algebraic group on elements outside the connected component of identity to character of an irreducible representation of another connected semisimple algebraic group can be deduced from properties of the canonical bases.
The extended weight semigroup of a homogeneous space G/H of a connected semisimple algebraic group G characterizes the spectra of the representations of G on the spaces of regular sections of homogeneous linear bundles over G/H, including the space of regular functions on G/H. We compute the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces G/H, where G is a simply connected non-simple semisimple complex algebraic group and H a connected closed subgroup of it. In all the cases we also find the highest weight functions corresponding to the indecomposabl…
