A basis for the symplectic group branching algebra
Sangjib Kim
and Oded Yacobi
DOI: 10.1007/s10801-011-0303-7
Abstract
The symplectic group branching algebra, B \mathcal {B}, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp 2 n - 2(\Bbb C) in each finite-dimensional irreducible representation of Sp 2 n (\Bbb C). By describing on B \mathcal {B} an ASL structure, we construct an explicit standard monomial basis of B \mathcal {B} consisting of Sp 2 n - 2(\Bbb C) highest weight vectors. Moreover, B \mathcal {B} is known to carry a canonical action of the n-fold product SL 2\times ... \times SL 2, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi algebras we describe a deformation of Spec( B) \mathrm{Spec}(\mathcal {B}) into an explicitly described toric variety.
Pages: 269–290
Keywords: keywords symplectic groups; branching rules; hibi algebra; algebra with straightening law
Full Text: PDF
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2. Conca, A., Herzog, J., Valla, G.: Sagbi bases with applications to blow-up algebras. J. Reine Angew. Math. 474, 113-138 (1996)
3. De Concini, C., Eisenbud, D., Procesi, C.: Hodge Algebras. Astéisque, vol.
91. Société Mathématique de France, Paris (1982). 87 pp.
4. Eisenbud, D.: Introduction to algebras with straightening laws. In: Ring Theory and Algebra, III, Proc. Third Conf., Univ. Oklahoma, Norman, Okla.,
1979. Lecture Notes in Pure and Appl. Math., vol. 55, pp. 243-268. Dekker, New York (1980)
5. Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants. Graduate Texts in Mathematics, vol.
255. Springer, Dordrecht (2009)
6. Hibi, T.: Distributive lattices, affine semigroup rings and algebras with straightening laws. In: Commutative Algebra and Combinatorics, Kyoto,
1985. Adv. Stud. Pure Math., vol. 11, pp. 93-109. North- Holland, Amsterdam (1987)
7. Howe, R.: Weyl Chambers and standard monomial theory for poset lattice cones. Q. J. Pure Appl. Math. 1(1), 227-239 (2005)
8. King, R.C.: Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups. J. Math. Phys. 12, 1588-1598 (1971)
9. King, R.C.: Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8, 429-449 (1975)
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11. Koike, K., Terada, I.: Young-diagrammatic methods for the representation theory of the classical groups of type Bn, Cn, Dn. J. Algebra 107(2), 466-511 (1987) J Algebr Comb (2012) 35:269-290
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