A plactic algebra of extremal weight crystals and the Cauchy identity for Schur operators
Jae-Hoon Kwon
DOI: 10.1007/s10801-011-0278-4
Abstract
We give a new bijective interpretation of the Cauchy identity for Schur operators which is a commutation relation between two formal power series with operator coefficients. We introduce a plactic algebra associated with the Kashiwara's extremal weight crystals over the Kac-Moody algebra of type A +\infty , and construct a Knuth type correspondence preserving the plactic relations. This bijection yields the Cauchy identity for Schur operators as a homomorphic image of its associated identity for plactic characters of extremal weight crystals, and also recovers Sagan and Stanley's correspondence for skew tableaux as its restriction.
Pages: 427–449
Keywords: keywords plactic algebra; crystal; Schur operator
Full Text: PDF
References
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3. Fomin, S.: Schur operators and Knuth correspondences. J. Comb. Theory, Ser. A 72, 277-292 (1995)
4. Fomin, S., Greene, C.: Noncommutative Schur functions and their applications. Discrete Math. 193(1- 3), 179-200 (1998)
5. Frenkel, I.B.: Representations of Kac-Moody algebras and dual resonance models. In: Applications of Group Theory in Physics and Mathematical Physics. Lectures in Appl. Math., vol. 21, pp. 325-353. AMS, Providence (1985)
6. Fulton, W.: Young Tableaux. London Mathematical Society Student Texts, vol.
35. Cambridge University Press, Cambridge (1997)
7. Kashiwara, M.: On crystal bases. In: Representations of Groups. CMS Conf. Proc., vol. 16, pp. 155-
197. Amer. Math. Soc., Providence (1995)
8. Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J. 73, 383-413 (1994)
9. Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165, 295-345 (1994)
10. Knuth, D.: Permutations, matrices, and the generalized Young tableaux. Pac. J. Math. 34, 709-727 (1970)
11. Kwon, J.-H.: Differential operators and crystals of extremal weight modules. Adv. Math. 222, 1339- 1369 (2009)
12. Kwon, J.-H.: Crystal bases of modified quantized enveloping algebras and a double RSK correspondence. Preprint (2010), . J. Comb. Theory Ser. A (to appear)
13. Lam, T.: Ribbon Schur operators. Eur. J. Comb. 29(1), 343-359 (2008)
14. Lascoux, A., Schützenberger, M.P.: Le monoïde plaxique. In: Noncommutative Structures in Algebra and Geometric Combinatorics, Naples,
1978. Quad. “Ricerca Sci.”, vol. 109, pp. 129-156. CNR, Rome (1981)
15. Littelmann, P.: A plactic algebra for semisimple Lie algebras. Adv. Math. 124, 312-331 (1996)
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