On 321-Avoiding Permutations in Affine Weyl Groups
R.M. Green
DOI: 10.1023/A:1015012524524
Abstract
We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type A n - 1 by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in W coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of W (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations.
Using Shi's characterization of the Kazhdan-Lusztig cells in the group W, we use our main result to show that the fully commutative elements of W form a union of Kazhdan-Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan-Lusztig basis of the associated Hecke algebra to be computed combinatorially.
Pages: 241–252
Keywords: pattern avoidance; Kazhdan-Lusztig cells
Full Text: PDF
References
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2. S.C. Billey and G.S. Warrington, “Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations,” J. Alg. Combin. 13 (2001), 111-136.
3. H. Eriksson and K. Eriksson, “Affine Weyl groups as infinite permutations,” Electronic J. Combin. 5 (1998), no. 1, Research Paper 18, 32pp. (electronic).
4. C.K. Fan, “Structure of a Hecke algebra quotient,” J. Amer. Math. Soc. 10 (1997), 139-167.
5. C.K. Fan and R.M. Green, “On the affine Temperley-Lieb algebras,” Jour. L.M.S. 60 (1999), 366-380.
6. C.K. Fan and J.R. Stembridge, “Nilpotent orbits and commutative elements,” J. Alg. 196 (1997), 490-498.
7. R.M. Green, “The affine q-Schur algebra,” J. Alg. 215 (1999), 379-411.
8. R.M. Green, “Tabular algebras and their asymptotic versions,” J. Algebra, to appear; math.QA/0107230.
9. R.M. Green and J. Losonczy, “Canonical bases for Hecke algebra quotients,” Math. Res. Lett. 6 (1999), 213-222.
10. R.M. Green and J. Losonczy, “Fully commutative Kazhdan-Lusztig cells,” Ann. Inst. Fourier (Grenoble) 51 (2001), 1025-1045.
11. C. Greene and D.J. Kleitman, “The structure of Sperner k-families,” J. Combin. Theory Ser. A 20 (1976), 41-68.
12. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990.
13. D. Kazhdan and G. Lusztig, “Representations of Coxeter groups and Hecke algebras,” Invent. Math. 53 (1979), 165-184.
14. G. Lusztig, “Some examples of square integrable representations of semisimple p-adic groups,” Trans. Amer. Math. Soc. 277 (1983), 623-653.
15. P. Papi, “Inversion tables and minimal left coset representatives for Weyl groups of classical type,” J. Pure Appl. Alg. 161 (2001), 219-234.
16. J.Y. Shi, “The Kazhdan-Lusztig cells in certain affine Weyl groups,” Lecture Notes in Mathematics, Vol. 1179, Springer, Berlin, 1986.
17. J.Y. Shi, “The partial order on two-sided cells of certain affine Weyl groups,” J. Alg. 176 (1996), 607-621.
18. J.R. Stembridge, “On the fully commutative elements of Coxeter groups,” J. Alg. Combin. 5 (1996), 353-385.
19. N. Xi, “The based ring of two-sided cells of affine Weyl groups of type Ãn - 1,” to appear; math.QA/0010159.
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