Crystal graphs for Lie superalgebras and Cauchy decomposition
Jae-Hoon Kwon
University of Seoul Department of Mathematics 90 Cheonnong-dong, Dongdaemun-gu Seoul 130-743 Korea 90 Cheonnong-dong, Dongdaemun-gu Seoul 130-743 Korea
DOI: 10.1007/s10801-006-0024-5
Abstract
We discuss Cauchy type decompositions of crystal graphs for general linear Lie superalgebras. More precisely, we consider bicrystal graph structures on various sets of matrices of non-negative integers, and obtain their decompositions with explicit combinatorial isomorphisms.
Pages: 57–100
Keywords: keywords crystal graphs; Lie superalgebra; Cauchy decomposition
Full Text: PDF
References
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2. A. Berele and A. Regev, “Hook Young diagrams with applications to combinatorics and to the representations of Lie superalgebras,” Adv. Math. 64 (1987), 118-175.
3. S.-J. Cheng and N. Lam, “Infinite-dimensional Lie superalgebras and hook Schur functions,” Comm. Math. Phys. 238(1-2) (2003), 95-118.
4. S.-J. Cheng and W. Wang, “Howe duality for Lie superalgebras,” Compositio Math. 128(1) (2001), 55-94.
5. S.-J. Cheng and W. Wang, “Lie subalgebras of differential operators on the super circle,” Publ. Res. Inst. Math. Sci. 39(3) (2003), 545-600.
6. V.I. Danilov and G.A. Koshevoy, “Bi-crystals and crystal (G L(V ), G L(W )) duality,” RIMS preprint, (2004) No. 1458.
7. W. Fulton, “Young tableaux,” London Mathematical Society Student Texts, Vol. 35 Cambridge University Press,” Cambridge, 1997.
8. K. Hasegawa, “Spin module versions of Weyl's reciprocity theorem for classical Kac-Moody Lie algebras-an application to branching rule duality,” Publ. Res. Inst. Math. Sci. 25(5) (1989), 741-828.
9. R. Howe, “Remarks on classical invariant theory,” Trans. Amer. Math. Soc. 313(2) (1989), 539-570.
10. V.G. Kac, Infinite Dimensional Lie Algebras. Cambridge University Press, 3rd ed., 1990.
11. V.G. Kac, “Lie superalgebras,” Adv. in Math. 26(1) (1977), 8-96.
12. V.G. Kac and J.W. van de Leur, “Super boson-fermion correspondence,” Ann. Inst. Fourier 37(4) (1987), 99-137.
13. V. G. Kac and A. Radul, “Representation theory of the vertex algebra W1+\infty ,” Transform. Groups 1(1-2) (1996), 41-70.
14. S.-J. Kang and J.-H. Kwon, “Tensor product of crystal bases for Uq (gl(m, n))-modules,” Comm. Math. Phys. 224 (2001), 705-732.
15. M. Kashiwara, “Crystal bases and Littelmann's refined Demazure character formula,” Duke Math. J. 71 (1993), 839-858.
16. M. Kashiwara, “On crystal bases,” Representations of groups, in CMS Conf. Proc., Vol. 16, Amer. Math. Soc., Providence, RI, (1995), pp. 155-197.
17. D. Knuth, “Permutations, matrices, and the generalized Young tableaux,” Pacific J. Math. 34 (1970), 709-727.
18. I.G. Macdonald, “Symmetric functions and Hall polynomials,” Oxford University Press, 2nd ed., 1995.
19. J.B. Remmel, “The combinatorics of (k, l)-hook Schur functions,” Contemp. Math. 34 (1984), 253-287.
20. J.R. Stembridge, “Rational tableaux and the tensor algebra of gln,” J. Combin. Theory Ser. A 46(1) (1987), 79-120.