Total Nonnegativity and Induced Sign Characters of the Hecke Algebra

Abstract

Let \({\mathfrak {S}}_{[i,j]}\) be the subgroup of the symmetric group \({\mathfrak {S}}_n\) generated by adjacent transpositions \((i,i+1), \dotsc , (j-1,j)\), assuming \(1 \le i < j \le n\). We give a combinatorial rule for evaluating induced sign characters of the type A Hecke algebra \(H_n(q)\) at all elements of the form \(\sum _{w \in {\mathfrak {S}}_{[i,j]}} T_w\) and at all products of such elements. This includes evaluation at some elements \(C'_w(q)\) of the Kazhdan–Lusztig basis.

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References

  1. 1.

    A. A. Beĭlinson, J. Bernstein, and P. Deligne. Faisceaux pervers. In Analysis and topology on singular spaces, I (Luminy, 1981), vol. 100 of Astérisque. Soc. Math. France, Paris (1982), pp. 5–171.

  2. 2.

    S. C. Billey and G. Warrington. Kazhdan-Lusztig polynomials for \(321\)-hexagon-avoiding permutations. J. Algebraic Combin., 13, 2 (2001) pp. 111–136.

    MathSciNet  Article  Google Scholar 

  3. 3.

    A. Björner and F. Brenti. Combinatorics of Coxeter groups, vol. 231 of Graduate Texts in Mathmatics. Springer, New York (2005).

  4. 4.

    F. Brenti. Combinatorics and total positivity. J. Combin. Theory Ser. A, 71, 2 (1995) pp. 175–218.

    MathSciNet  Article  Google Scholar 

  5. 5.

    S. Clearman, M. Hyatt, B. Shelton, and M. Skandera. Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements. Electron. J. Combin., 23, 2 (2016). Paper 2.7, 56 pages.

  6. 6.

    C. W. Cryer. Some properties of totally positive matrices. Linear Algebra Appl., 15 (1976) pp. 1–25.

    MathSciNet  Article  Google Scholar 

  7. 7.

    V. Deodhar. A combinatorial setting for questions in Kazhdan-Lusztig theory. Geom. Dedicata, 36, 1 (1990) pp. 95–119.

    MathSciNet  Article  Google Scholar 

  8. 8.

    S. Fomin and A. Zelevinsky. Total positivity: Tests and parametrizations. Math. Intelligencer, 22, 1 (2000) pp. 23–33.

    MathSciNet  Article  Google Scholar 

  9. 9.

    F. R. Gantmacher and M. G. Krein. Oscillation matrices and kernels and small vibrations of mechanical systems. AMS Chelsea Publishing, Providence (2002). Edited by A. Eremenko. Translation based on the 1941 Russian original.

  10. 10.

    M. Haiman. Hecke algebra characters and immanant conjectures. J. Amer. Math. Soc., 6, 3 (1993) pp. 569–595.

    MathSciNet  Article  Google Scholar 

  11. 11.

    R. Kaliszewski, J. Lambright, and M. Skandera. Bases of the quantum matrix bialgebra and induced sign characters of the Hecke algebra. J. Algebraic Combin., 49, 4 (2019) pp. 475–505.

    MathSciNet  Article  Google Scholar 

  12. 12.

    S. Karlin and G. McGregor. Coincidence probabilities. Pacific J. Math., 9 (1959) pp. 1141–1164.

    MathSciNet  Article  Google Scholar 

  13. 13.

    D. Kazhdan and G. Lusztig. Representations of Coxeter groups and Hecke algebras. Invent. Math., 53 (1979) pp. 165–184.

    MathSciNet  Article  Google Scholar 

  14. 14.

    M. Konvalinka and M. Skandera. Generating functions for Hecke algebra characters. Canad. J. Math., 63, 2 (2011) pp. 413–435.

    MathSciNet  Article  Google Scholar 

  15. 15.

    V. Lakshmibai and B. Sandhya. Criterion for smoothness of Schubert varieties in \(SL(n)/B\). Proc. Indian Acad. Sci. (Math Sci.), 100, 1 (1990) pp. 45–52.

    MathSciNet  Article  Google Scholar 

  16. 16.

    B. Lindström. On the vector representations of induced matroids. Bull. London Math. Soc., 5 (1973) pp. 85–90.

    MathSciNet  Article  Google Scholar 

  17. 17.

    G. Lusztig. Total positivity in reductive groups. In Lie Theory and Geometry: in Honor of Bertram Kostant, vol. 123 of Progress in Mathematics. Birkhäuser, Boston (1994), pp. 531–568.

  18. 18.

    Y. I. Manin. Quantum groups and noncommutative geometry. Université de Montréal Centre de Recherches Mathématiques, Montreal, QC (1988).

  19. 19.

    M. Skandera. On the dual canonical and Kazhdan-Lusztig bases and 3412, 4231-avoiding permutations. J. Pure Appl. Algebra, 212 (2008).

  20. 20.

    T. A. Springer. Quelques aplications de la cohomologie d’intersection. In Séminaire Bourbaki, Vol. 1981/1982, vol. 92 of Astérisque. Soc. Math. France, Paris (1982), pp. 249–273.

  21. 21.

    J. Stembridge. Immanants of totally positive matrices are nonnegative. Bull. London Math. Soc., 23 (1991) pp. 422–428.

    MathSciNet  Article  Google Scholar 

  22. 22.

    J. Stembridge. Some conjectures for immanants. Canad. J. Math., 44, 5 (1992) pp. 1079–1099.

    MathSciNet  Article  Google Scholar 

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Correspondence to Mark Skandera.

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Clearwater, A., Skandera, M. Total Nonnegativity and Induced Sign Characters of the Hecke Algebra. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00545-4

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