We study branching multiplicity spaces of complex classical groups in terms of ${\mathrm{GL} }_{2} $ representations. In particular, we show how combinatorics of ${\mathrm{GL} }_{2} $ representations are intertwined to make branching rules under the restriction of ${\mathrm{GL} }_{n} $ to ${\mathrm{GL} }_{n- 2} $. We also discuss analogous results for the symplectic and orthogonal groups.

Let G be a complex connected reductive group. The Parthasarathy–Ranga Rao–Varadarajan (PRV) conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for ‘the same reason’ as the PRV ones.

We decompose the restriction of ramified principal series representations of the $p$-adic group $\text{GL}\left( 3,\,\text{k} \right)$ to its maximal compact subgroup $K\,=\,\text{GL}\left( 3,\,\mathcal{R} \right)$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples.

We construct bases for the stable branching algebras for the symmetric pairs $(\mathrm{GL}_{2n},\mathrm{Sp}_{2n}),\ (\mathrm{Sp}_{2(n+m)}, \mathrm{Sp}_{2n}\times\mathrm{Sp}_{2m})$ and $(\mathrm{O}_{2n},\mathrm{GL}_{n})$. Each basis element is expressed as a sum of products of pfaffians.

Let $F$ be an algebraically closed field of characteristic $p$, and let $H$ be a finite group. A natural and important problem in representation theory is to classify the pairs $(G, D)$, where $G$ is a subgroup of $H$, and $D$ is an irreducible $FH$-module that remains irreducible when restricted to $G$. For example, if $H$ is an almost simple group or a central extension of an almost simple group, then one encounters this problem in the attempt to determine the maximal subgroups of the finite classical groups. We solve the above problem for Schur's double covers $H = {\widehat S}_n$ and ${\widehat A}_n$ of the symmetric and alternating groups, and $G$ a Young-type subgroup. The answer is given in terms of the combinatorics of restricted, $p$-strict partitions; such partitions are used to parameterize the irreducible spin representations of ${\widehat S}_n$ and ${\widehat A}_n$. We exploit the connections, recently developed by Ariki, Grojnowski, Brundan, Kleshchev, and others, between modular branching rules, crystal graphs of affine Kac–Moody algebras, and affine Hecke algebras and related objects.

We prove the quantum version - for Hecke algebras $H(A_n)$ of type $A$ at roots of unity - of Kleshchev's modular branching rule for symmetric groups. This result describes the socle of the restriction of an irreducible $H(A_n)$-module to the subalgebra $H(A_{n-1})$. As a consequence, we describe the quantum version of the Mullineux involution describing the irreducible module obtained on twisting an irreducible module with the sign representation.

1991 Mathematics Subject Classification: 20C05, 20G05.