Compression of Nakajima monomials in type A and C
Matthias Meng
DOI: 10.1007/s10801-011-0319-z
Abstract
We describe an explicit crystal morphism between Nakajima monomials and monomials which give a realization of crystal bases for finite dimensional irreducible modules over the quantized enveloping algebra for Lie algebras of type A and C. This morphism provides a connection between arbitrary Nakajima monomials and Kashiwara-Nakashima tableaux. This yields a translation of Nakajima monomials to the Littelmann path model. Furthermore, as an application of our results we define an insertion scheme for Nakajima monomials compatible to the insertion scheme for tableaux.
Pages: 649–690
Keywords: keywords crystals; nakajima monomials; littelmann paths; insertion scheme; tableaux
Full Text: PDF
References
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2. Kang, S.-J., Kim, S.-J., Shin, D.-U.: Monomial realization of crystal bases for special linear Lie algebras. J. Algebra 274, 629-642 (2004)
3. Kang, S.-J., Kim, J.-A., Shin, D.-U.: Crystal bases for quantum classical algebras and Nakajima's monomials. Publ. Res. Inst. Math. Sci. 40, 758-791 (2004)
4. Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63, 465-516 (1991)
5. Kashiwara, M.: Realizations of crystals. Contemp. Math. 325, 133-139 (2003)
6. Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165, 295-345 (1994)
7. Kim, J.-A., Shin, D.-U.: Correspondence between Young walls and Young tableaux realizations of crystal bases for the classical Lie algebras. J. Algebra 282, 728-757 (2004)
8. Kim, J.-A., Shin, D.-U.: Insertion schemes for the classical Lie algebras. Commun. Algebra 32, 3139- 3167 (2004)
9. Littelmann, P.: Paths and root operators in representation theory. Ann. Math. 142, 449-525 (1995)
10. Littelmann, P.: Characters of representations and paths in * hR. Proc. Symp. Pure Math. 61, 29-49 (1997)
11. Nakajima, H.: t -Analogs of q-characters of quantum affine algebras of type An and Dn. Contemp.
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