Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem

Abstract

Abstract. The Kronecker coefficient g λµν is the multiplicity of the GL(V ) × GL(W )-irreducible V λ ⊗ W µ in the restriction of the GL(X)-irreducible X ν via the natural map, where V, W are C-vector spaces and X = V ⊗ W . A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients.We construct two quantum objects for this problem, which we call the nonstandard quantum group and nonstandard Hecke algebra. We show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality.Using these nonstandard objects as a guide, we follow the approach of Adsul, Sohoni, and Subrahmanyam [1] to construct, in the case dim(V ) = dim(W ) = 2, a representatioň X ν of the nonstandard quantum group that specializes to Res GL(V )×GL(W ) X ν at q = 1. We then define a global crystal basis +HNSTC(ν) ofX ν that solves the two-row Kronecker problem: the number of highest weight elements of +HNSTC(ν) of weight (λ, µ) is the Kronecker coefficient g λµν . We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the U q (sl 2 ) graphical calculus from [19], and use this to organize the crystal components of +HNSTC(ν) into eight families. This yields a fairly simple, positive formula for two-row Kronecker coefficients, generalizing a formula in [15]. As a byproduct of the approach, we also obtain a rule for the decomposition of Res GL2×GL2⋊S2 X ν into irreducibles.

Kronecker coefficients and noncommutative super Schur functions

“…This amounts to showing that if w and w ′ differ by a single application of one of the relations (4)- (7), (9), (25), then w ∈ CYW λ,d ⇐⇒ w ′ ∈ CYW λ,d . Since CYW λ,d is shuffle closed, the necessary result holds for w and w ′ that differ by (25) or by (4)- (7), (9) in the case that exactly one of the two letters involved is barred. Since Knuth transformations preserve whether an ordinary word is Yamanouchi, the necessary result holds if w and w ′ differ by (4)- (7) and the two letters are both barred or both unbarred.…”

“…Since Knuth transformations preserve whether an ordinary word is Yamanouchi, the necessary result holds if w and w ′ differ by (4)- (7) and the two letters are both barred or both unbarred. Finally, an ordinary word's being Yamanouchi is unchanged by swapping two adjacent letters that differ by more than 1, which handles the case that w and w ′ differ by (9) and the two letters involved are both barred or both unbarred.…”

Geometric complexity theory V: Efficient algorithms for Noether normalization

“…This article belongs to a series [71,73,70,6] of articles on geometric complexity theory. See [67,66] for an overview of the earlier articles in this series, and [14] for an overview of the mathematical issues therein.…”

Unifying Known Lower Bounds via Geometric Complexity Theory

“…One advantage of considering multiplicities rather than test modules is that it opens the possibility of using purely representation-theoretic techniques to understand the multiplicities, as is being pursued in GCT (e. g., [34], [35], [36], [37]). To see how this is possiblethat is, how one can discuss multiplicity obstructions without reference to actual test polynomials or modules thereof-we must mention a bit more about the representation theory of GL n and S n .…”