A Hopf algebraic approach to the theory of group branch- ings

We describe a Hopf algebraic approach to the Grothendieck ring of representations of subgroups Hπ of the general linear group GL(n) which stabilize a tensor of Young symmetry {π}. It turns out that the representation ring of the subgroup can be described as a Hopf algebra twist, with a 2-cocycle derived from the Cauchy kernel 2-cocycle using plethysms. Due to Schur-Weyl duality we also need to employ the coproduct of the inner multiplication. A detailed analysis including combinatorial proofs for our results can be found in [4]. In this paper we focus on the Hopf algebraic treatment, and a more formal approach to representation rings and symmetric functions. 1. Group representation rings We are interested in the representation rings of GL(n) and its subgroups described as stabilizers of certain elements T π of Young symmetry π. A matrix representation ρ : GL(n) → GL(m), m ≥ n is polynomial if the entries of ρ(g) ∈ GL(m) are polynomials in the entries of g ∈ GL(n). The character of the representation ρ is the central function χρ : GL(n) → C, χρ(g) = tr(ρ(g)). Representations form an Abelian semigroup under the direct sum V λ ⊕ V μ, which is completed to form the Grothendieck group Rn = RGL(n)({V λ},⊕) using virtual representations −V λ [9]. The tensor product V λ ⊗ V μ = ⊕νCν λμV ν turns this structure into a ring Rn = RGL(n)({V λ},⊕,⊗). We proceed to the B. Fauser, P.D. Jarvis, R.C.King : A Hopf algebraic approach to group branchings 2 inductive limit RGL = lim←R n since finitely generated representation rings develop syzygies while the limit ring is free. Finite examples thus require establishing these syzygies by so-called modification rules. We follow [15]. GL(n) acts by conjugation on the Lie algebra gl(n) of all n × n-matrices, hence acts on the invariant ring Pol(gl(n))GL(n) with integer coefficients (in Lie theory real coefficients). Under some topological restrictions one can identify the characters χ ∈ R bijectively with elements in Pol(gl(n))GL(n) via the isomorphism φ : RGL(n) → Pol(gl(n))GL(n) of class functions. A particular basis of the representation ring is given by equivalence classes of irreducible representations V λ, labelled by integer partitions λ (see below). For each partition label λ there exists a Schur map (Schur endofunctor on FinVectC, [11]) mapping the vector space V = C n the corresponding Schur module V λ, a highest weight GL(n)-module in Lie theory. The character of V λ is the Schur polynomial sλ that is polynomial in the eigenvalues of g ∈ GL(n). Let dimV = n. On any tensor product W = ⊗pV we have the left GL(n) action and right action of the symmetric group Sp. As bimodule we have W = ⊗pV = ⊕λV λ ⊗ Sλ by Schur-Weyl duality [17, 6]. One considers Young symmetrizers Yλ = cλrλ where cλ is the column antisymmetrizer of the tableau of shape λ and rλ is the row symmetrizer. The Yλ are idempotents and reduce W into irreducible parts with respect to GL(n)× Sp. The Schur module V λ is the image of the identity morphisms W → W defined as right multiplication by Yλ. This twofold nature of Schur polynomials will cause the remarkable self duality of the Hopf algebra studied in the next paragraph. 2. Symmetric functions 2.1. The Hopf algebra of symmetric functions An introduction to symmetric functions can be found e.g. in [12, 7, 11], the well known Hopf algebra structure is discussed in [5, 19, 16, 8]. Here we focus on the Hopf algebraic aspects of symmetric functions related to representation rings of the GL groups and their Weyl groups lim←⊕Sp using the isomorphism between GL(n) and Sp representation rings and Λ = Z[x1, x2, . . .] S = lim←⊕ n Z[x1, . . . , xn] S of symmetric functions in infinitely many variables. Schur functions sλ, or in Littlewood’s bracket notation {λ}, are indexed by integer partitions λ = (λ1, . . . , λk) = [1 r1 , . . . , prp ], where the B. Fauser, P.D. Jarvis, R.C.King : A Hopf algebraic approach to group branchings 3 λi, ordered by magnitude λi ≥ λi+1, are called parts, the ri are multiplicities, and are conveniently displayed by Ferrers diagrams (also called Young diagrams). Schur functions are given by sλ(x) = ∑ T∈STλ x wgt (T ), where the sum is over all tableaux (fillings) T belonging to the set ST λ of semi-standard tableaux (column strict, row semistrict) of shape λ. Each summand is a monomial in the variables x1, x2, . . . , xn of degree n = |λ| = ∑ λi. The module underlying Λ is spanned by Z-linear combinations of Schur functions (irreducible representations). To establish the ring structure we introduce the outer multiplication V λ ⊗ V μ = ⊕C λμV ν φ ⇔ sλ(x) · sμ(x) = ∑ ν C λμsν(x). (1) Where the nonnegative integer constants Cν λμ are the famous LittlewoodRichardson coefficients determined e.g. combinatorially. Schur functions are important because they encode characters of irreducible representations of the GL(n) groups which by Schur’s lemma decompose into isoclasses. The Schur-Hall scalar product encodes this fact letting Schur functions be orthogonal by definition 〈 | 〉 : Λ⊗ZΛ → Λ, 〈sλ | sμ〉 = δλ,μ. This implies an elementwise identification of the module underlying Λ with the dual module Λ⋆ = Hom(Λ,Z). Λ⋆ is a priori not an algebra! However, inspection of classical results shows [3] that we can introduce the same outer product on the dual Λ⋆ reflecting the Frobenius reciprocity. Using the Milnor-Moore theorem this induces a coalgebraic structure on Λ fulfilling the axioms of a Hopf algebra [3]. Schur functions sλ have a life as characters of GL(n)-modules V λ and through the Schur-Weyl duality mentioned previously are also associated with irreducible representations of Sp, a remarkable incidence. With f, h, g ∈ Λ we define the outer coproduct ∆ : Λ → Λ⊗ Λ as 〈∆(f)|g ⊗ h〉 := 〈f |g · h〉 = 〈f(1)|g〉〈f(2)|h〉 ∆(sλ) = ∑