W-graph versions of tensoring with the S n \mathcal{S}_{n} defining representation
Jonah Blasiak
DOI: 10.1007/s10801-011-0281-9
Abstract
We further develop the theory of inducing W-graphs worked out by Howlett and Yin (Math. Z. 244(2):415-431, 2003 and Manuscr. Math. 115(4):495-511, 2004), focusing on the case W = S n W = \mathcal{S}_{n}. Our main application is to give two W-graph versions of tensoring with the S n \mathcal{S}_{n} defining representation V, one being S n, S n -1 \mathcal{S}_{n}, \mathcal{S}_{n-1} and the other 
, where 
is a subalgebra of the extended affine Hecke algebra and the subscript signifies taking the degree 1 part. We look at the corresponding W-graph versions of the projection V\otimes V\otimes - \rightarrow S 2 V\otimes - . This does not send canonical basis elements to canonical basis elements, but we show that it approximates doing so as the Hecke algebra parameter u\rightarrow 0. We make this approximation combinatorially explicit by determining it on cells and relate this to RSK growth diagrams.
, where is a subalgebra of the extended affine Hecke algebra and the subscript signifies taking the degree 1 part. We look at the corresponding W-graph versions of the projection V\otimes V\otimes - \rightarrow S 2 V\otimes - . This does not send canonical basis elements to canonical basis elements, but we show that it approximates doing so as the Hecke algebra parameter u\rightarrow 0. We make this approximation combinatorially explicit by determining it on cells and relate this to RSK growth diagrams.Pages: 545–585
Keywords: keywords W-graph; Hecke algebra; restriction and induction; canonical basis; growth diagram
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References
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1991. Proc. Sympos. Pure Math., vol. 56, pp. 135-148. Am. Math. Soc., Providence (1994)
3. Fomin, S.: Knuth equivalence, jeu de taquin, and the Littlewood-Richardson rule. In: Appendix I in Enumerative Combinatorics, Vol.
2. Cambridge Studies in Advanced Mathematics, vol.
62. Cambridge University Press, Cambridge (1999)
4. Geck, M.: On the induction of Kazhdan-Lusztig cells. Bull. Lond. Math. Soc. 35(5), 608-614 (2003)
5. Geck, M.: Relative Kazhdan-Lusztig cells. Represent. Theory 10, 481-524 (2006) (electronic)
6. Graham, J.J., Lehrer, G.I.: Cellular algebras. Invent. Math. 123(1), 1-34 (1996)
7. Haiman, M.: Cherednik algebras, Macdonald polynomials and combinatorics. In: International Congress of Mathematicians, vol. III, pp. 843-872. Eur. Math. Soc., Zürich (2006)
8. Howlett, R.B., Yin, Y.: Inducing W -graphs. Math. Z. 244(2), 415-431 (2003)
9. Howlett, R.B., Yin, Y.: Inducing W -graphs. II. Manuscr. Math. 115(4), 495-511 (2004)
10. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165-184 (1979)
11. Lusztig, G.: Cells in affine Weyl groups. II. J. Algebra 109(2), 536-548 (1987)
12. Roichman, Y.: Induction and restriction of Kazhdan-Lusztig cells. Adv. Math. 134(2), 384-398 (1998)
13. Stanley, R.P.: In: Enumerative Combinatorics, vol.
2. Cambridge Studies in Advanced Mathematics, vol.
62. Cambridge University Press, Cambridge (1999)
14. Xi, N.: The Based Ring of Two-Sided Cells of Affine Weyl Groups of Type An -
1. Mem. Am. Math.
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