Towards combinatorial invariance for Kazhdan-Lusztig polynomials. (English) Zbl 1509.20057
With every pair of elements in a Coxeter group, a polynomial with integer coefficients is associated which is known as the Kazhdan-Lusztig polynomial, and it was introduced in the fundamental paper [D. Kazhdan and G. Lusztig, Invent. Math. 53, 165–184 (1979; Zbl 0499.20035)]. In that paper, the nonnegativity of the coefficients of the polynomial was conjectured, and this was proved in, e.g. [B. Elias and G. Williamson, Ann. Math. (2) 180, No. 3, 1089–1136 (2014; Zbl 1326.20005)]. The Kazhdan-Lusztig polynomials codify important information to understand the Bruhat order of Coxeter groups and the geometry of Schubert varietes.
Lusztig (1983) and Dyer (1987) conjectured that the K-L polynomial associated with the two elements \(x\) and \(y\), depends only of the Bruhat inverval \([x,y]\); this is known as the combinatorial invariance conjecture. In the present work, the authors provide a new combinatorial formula for the K-L polynomials of symmetric groups. Their formula suggests a feasible approach to the combinatorial invariance conjecture for symmetric groups. Along the paper, they introduce the notion of an hypercube decomposition and propose an interesting conjecture related to this new concept.
Recommended introductions are [B. Elias et al., Introduction to Soergel bimodules. Cham: Springer (2020; Zbl 1507.20001); N. Xi, Sci. Sin., Math. 47, No. 11, 1467–1480 (2017; Zbl 1499.20005); F. Brenti, Sémin. Lothar. Comb. 49, B49b, 30 p. (2002; Zbl 1036.20037)]. See also [N. Libedinsky and G. Williamson, J. Algebra 568, 181–192 (2021; Zbl 1458.20036); D. Bump and M. Nakasuji, Can. J. Math. 71, No. 6, 1351–1366 (2019; Zbl 1459.22004); N. Proudfoot, EMS Surv. Math. Sci. 5, No. 1–2, 99–127 (2018; Zbl 1445.14038); M. Marietti, Trans. Am. Math. Soc. 368, No. 7, 5247–5269 (2016; Zbl 1331.05232); B. Elias and G. Williamson, Prog. Math. 319, 105–126 (2016; Zbl 1367.20002); A. Björner and T. Ekedahl, Ann. Math. (2) 170, No. 2, 799–817 (2009; Zbl 1226.05268); J. Brundan, J. Algebra 306, No. 1, 17–46 (2006; Zbl 1169.17008); F. Caselli, J. Algebr. Comb. 18, No. 3, 171–187 (2003; Zbl 1067.05078); F. Brenti and R. Simion, J. Algebr. Comb. 11, No. 3, 187–196 (2000; Zbl 0958.05138); P. Polo, Represent. Theory 3, 90–104 (1999; Zbl 0968.14029); F. Brenti, Discrete Math. 193, No. 1–3, 93–116 (1998; Zbl 1061.05511); Invent. Math. 118, No. 2, 371–394 (1994; Zbl 0836.20054); V. V. Deodhar, Contemp. Math. 88, 579–583 (1989; Zbl 0685.17011); B. Elias et al., Adv. Math. 299, 36–70 (2016; Zbl 1341.05250)].
Lusztig (1983) and Dyer (1987) conjectured that the K-L polynomial associated with the two elements \(x\) and \(y\), depends only of the Bruhat inverval \([x,y]\); this is known as the combinatorial invariance conjecture. In the present work, the authors provide a new combinatorial formula for the K-L polynomials of symmetric groups. Their formula suggests a feasible approach to the combinatorial invariance conjecture for symmetric groups. Along the paper, they introduce the notion of an hypercube decomposition and propose an interesting conjecture related to this new concept.
Recommended introductions are [B. Elias et al., Introduction to Soergel bimodules. Cham: Springer (2020; Zbl 1507.20001); N. Xi, Sci. Sin., Math. 47, No. 11, 1467–1480 (2017; Zbl 1499.20005); F. Brenti, Sémin. Lothar. Comb. 49, B49b, 30 p. (2002; Zbl 1036.20037)]. See also [N. Libedinsky and G. Williamson, J. Algebra 568, 181–192 (2021; Zbl 1458.20036); D. Bump and M. Nakasuji, Can. J. Math. 71, No. 6, 1351–1366 (2019; Zbl 1459.22004); N. Proudfoot, EMS Surv. Math. Sci. 5, No. 1–2, 99–127 (2018; Zbl 1445.14038); M. Marietti, Trans. Am. Math. Soc. 368, No. 7, 5247–5269 (2016; Zbl 1331.05232); B. Elias and G. Williamson, Prog. Math. 319, 105–126 (2016; Zbl 1367.20002); A. Björner and T. Ekedahl, Ann. Math. (2) 170, No. 2, 799–817 (2009; Zbl 1226.05268); J. Brundan, J. Algebra 306, No. 1, 17–46 (2006; Zbl 1169.17008); F. Caselli, J. Algebr. Comb. 18, No. 3, 171–187 (2003; Zbl 1067.05078); F. Brenti and R. Simion, J. Algebr. Comb. 11, No. 3, 187–196 (2000; Zbl 0958.05138); P. Polo, Represent. Theory 3, 90–104 (1999; Zbl 0968.14029); F. Brenti, Discrete Math. 193, No. 1–3, 93–116 (1998; Zbl 1061.05511); Invent. Math. 118, No. 2, 371–394 (1994; Zbl 0836.20054); V. V. Deodhar, Contemp. Math. 88, 579–583 (1989; Zbl 0685.17011); B. Elias et al., Adv. Math. 299, 36–70 (2016; Zbl 1341.05250)].
Reviewer: José Martínez-Bernal (Ciudad de México)
MSC:
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 20C08 | Hecke algebras and their representations |
| 05E10 | Combinatorial aspects of representation theory |
Keywords:
Bruhat graph; combinatorial invariance conjecture; Coxeter groups; hypercube decomposition; intersection cohomology; Kazhdan-Lusztig polynomials; Schubert varietiesCitations:
Zbl 0499.20035; Zbl 1326.20005; Zbl 1499.20005; Zbl 1036.20037; Zbl 1458.20036; Zbl 1459.22004; Zbl 1445.14038; Zbl 1331.05232; Zbl 1367.20002; Zbl 1226.05268; Zbl 1169.17008; Zbl 1067.05078; Zbl 0958.05138; Zbl 0968.14029; Zbl 1061.05511; Zbl 0836.20054; Zbl 0685.17011; Zbl 1341.05250; Zbl 1507.20001References:
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