A simplified Kronecker rule for one hook shape

Abstract

Abstract. Recently Blasiak gave a combinatorial rule for the Kronecker coefficient g λµν when µ is a hook shape by defining a set of colored Yamanouchi tableaux with cardinality g λµν in terms of a process called conversion. We give a characterization of colored Yamanouchi tableaux that does not rely on conversion, which leads to a simpler formulation and proof of the Kronecker rule for one hook shape.

Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

“…For partitions λ, µ, ν of nd let g(λ, µ, ν) ∈ N denote the Kronecker coefficient, i.e., the multiplicity of the irreducible S nd -representation [λ] in the tensor product [µ] ⊗ [ν], where [µ] ⊗ [ν] is interpreted as an S ndrepresentation via the diagonal embedding S nd ֒→ S nd × S nd , π → (π, π). A combinatorial interpretation of g(λ, µ, ν) is known only in special cases, see [Las80,Rem89,Rem92,RW94,Ros01,BO07,Bla12,Liu14,IMW15,Hay15], and finding a general combinatorial interpretation is problem 10 in Stanley's list of positivity problems and conjectures in algebraic combinatorics [Sta00]. In geometric complexity theory the main interest is focused on rectangular Kronecker coefficients, i.e., the coefficients g(λ, n × d, n × d).…”

“…In order to prove this theorem, we will derive a simple formula for these Kronecker coefficients, following the approaches set in [Bla12,Liu14,PP14b]. For brevity we set…”

“…For the sake of self-containment we repeat some calculations appearing in [Bla12,Liu14,PP14b]. We invoke Littlewood's identity, stating that…”

Tensor and matrix models: a one-night stand or a lifetime romance?

“…The most remarkable of them is perhaps the family of Kronecker coefficients with a hook shape. Started in [46] and refined in [47], this program succeeds in giving g µνλ a combinatorial interpretation when one of the partitions, say µ, is a hook shape. In this section we will use their results to built a compact formula of Kronecker coefficients with a hook shape in terms of Littlewood-Richardson numbers.…”

“…In [47] it was shown that the Kronecker coefficients g µ(r)νλ can be expressed in terms of the standard inner products 1 of Schur and skew Schur functions as…”