A combinatorial approach to classical representation theory

Abstract

A fundamental problem from invariant theory is to describe, given a representation V of a group G, the algebra End G (⊗ r V ) of multilinear functions invariant under the action of the group. According to Weyl's classic, a first main (later: 'fundamental') theorem of invariant theory provides a finite spanning set for this algebra, whereas a a second main theorem describes the linear relations between those basic invariants.Here we use diagrammatic methods to carry Weyl's programme a step further, providing explicit bases for the subspace of ⊗ r V invariant under the action of G, that are additionally preserved by the action of the long cycle of the symmetric group S r .The representations we study are essentially those that occur in Weyl's book, namely the defining representations of the symplectic groups, the defining representations of the symmetric groups considered as linear representations, and the adjoint representations of the linear groups.In particular, we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic groups Sp(2n). Our formulation is completely explicit and provides a very precise link to (n+1)-noncrossing perfect matchings, going beyond a dimension count. Extending our argument to the k-th symmetric powers of these representations, the combinatorial objects involved turn out to be (n + 1)-noncrossing k-regular graphs. As corollaries we obtain instances of the cyclic sieving phenomenon for these objects and the natural rotation action.In general, we are able to derive branching rules for the diagram algebras corresponding to the representations in a uniform way. Moreover, we compute the Frobenius characteristics of modules of the diagram algebras restricted to the action of the symmetric group. Via a general theorem, we also obtain the isotypic decomposition of ⊗ r V when n is large enough in comparison to r.

Partition algebras Pk(n) with 2k>n and the fundamental theorems of invariant theory for the symmetric group Sn

“…Generators for the kernels of these surjections give the Second Fundamental Theorem of Invariant Theory for the orthogonal and symplectic groups. As shown in the work of Hu and Xiao [HX], Lehrer and Zhang [LZ1,LZ2], and Rubey and Westbury [RW1], the kernels of these surjections are principally generated by a single idempotent when k ≥ n + 1. The recent work of Bowman, Enyang, and Goodman [BEG] adopts a cellular basis approach to describing the kernels in the orthogonal and symplectic cases, as well as in the case of the general linear group GL n acting on mixed tensor powers V ⊗k ⊗ (V * ) ⊗ℓ of its natural n-dimensional module V and its dual V * .…”

“…In [RW1,Sec. 7.4] (see also [RW2]), Rubey and Westbury consider the Brauer algebra B k (−2n), the related Brauer diagram category, and the commuting actions of B k (−2n) and the symplectic group Sp 2n afforded by the above surjection.…”

“…In [RW1,Sec. 8], Rubey and Westbury discuss the representation theory of the symmetric groups from the viewpoint of the partition category and the combinatorics of set partitions.…”

“…(n), with corresponding central idempotent, which they denote E ′ (k + 1). They conjecture (see [RW1,Conjecture 8.4.9]) that the idempotents E(k) and E ′ (k + 1) satisfy certain recurrence relations. We do not prove their recurrence relation conjecture, but rather in (6.33) and (6.35), we give exact expressions for E(k) and E ′ (k + 1) by showing that E(k) = Ξ k,n when n ≥ 2k and E ′ (k + 1) = Ξ k+ 1 2 ,n when n ≥ 2k + 1.…”

Combinatorics of symplectic invariant tensors

“…† We define pivotal and symmetric ideals in[14], and restrict ourselves here to down-to-earth language in Definitions 2.10 and 2.11.…”