2005/05/11 by Bertfried Fauser, B. Fauser, P. D. Jarvis +4
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Antisymmetric relation #Cellular algebra #Combinatorics #Invariant (physics) #Mathematics #Pure mathematics #Representation theory of the symmetric group #Ring of symmetric functions #Schur algebra #Symmetric function #Symmetric group #Tensor algebra #Tensor product #math-ph #math.MP #msc:05E05 #msc:11E57 #msc:16W30 #msc:20G10
paper · pdf · doi:10.1088/0305-4470/39/11/006
published as J. Phys. A: Math. Gen. 39 (2006) 2611-2655 · 40 pages, uses pstricks, dedicated to Brian G. Wybourne
arxiv created 2005/05/11 · openalex publication_date 2006/03/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings and fixing a vector v i , a symmetric tensor g ij = g ji and an antisymmetric tensor f ij = − f ji , respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function s π ≡ π by the basic M series of complete symmetric functions and the L = M −1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains π-generalized Newell–Littlewood formulae and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for and , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine and in some instances non-reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed.