Pattern avoidance and Boolean elements in the Bruhat order on involutions
Axel Hultman
and Kathrin Vorwerk
DOI: 10.1007/s10801-008-0152-1
Abstract
We show that the principal order ideal of an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin paths is demonstrated.
Pages: 87–102
Keywords: keywords Bruhat order; Boolean involutions; pattern avoidance
Full Text: PDF
References
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231. Springer, New York (2005)
3. Carrell, J.B.: The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties. In: Algebraic groups and their generalizations: classical methods, University Park, PA,
1991. Proc. Sympos. Pure Math., vol. 56, pp. 53-61. Amer. Math. Soc., Providence (1994) 4. du Cloux, F.: An abstract model for Bruhat intervals. European J. Combin. 21(2), 197-222 (2000)
5. Gasharov, V., Reiner, V.: Cohomology of smooth Schubert varieties in partial flag manifolds. J. London Math. Soc. (2) 66(3), 550-562 (2002) J Algebr Comb (2009) 30: 87-102
6. Hultman, A.: The combinatorics of twisted involutions in Coxeter groups. Trans. Amer. Math. Soc. 359, 2787-2798 (2007)
7. Hultman, A.: Twisted identities in Coxeter groups. J. Algebraic Combin. 28(2), 313-332 (2008)
8. Humphreys, J.E.: Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, vol.
29. Cambridge University Press, Cambridge (1990)
9. Incitti, F.: Bruhat order on the involutions of classical Weyl groups. Adv. in Appl. Math. 37(1), 68-111 (2006)
10. Lakshmibai, V., Sandhya, B.: Criterion for smoothness of Schubert varieties in SL(n)/B. Proc. Indian Acad. Sci. Math. Sci. 100(1), 45-52 (1990)
11. Richardson, R.W., Springer, T.A.: The Bruhat order on symmetric varieties. Geom. Dedicata 35(1-3), 389-436 (1990)
12. Sjöstrand, J.: Bruhat intervals as rooks on skew Ferrers boards. J. Combin. Theory Ser. A 114(7), 1182-1198 (2007)
13. Sloane, N.J.A.: The online encyclopedia of integer sequences.