Combinatorics of the immaculate inverse Kostka matrix
Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 1119-1142.

The classical Kostka matrix counts semistandard tableaux and expands Schur symmetric functions in terms of monomial symmetric functions. The entries in the inverse Kostka matrix can be computed by various algebraic and combinatorial formulas involving determinants, special rim hook tableaux, raising operators, and tournaments. Our goal here is to develop an analogous combinatorial theory for the inverse of the immaculate Kostka matrix. The immaculate Kostka matrix enumerates dual immaculate tableaux and gives a combinatorial definition of the dual immaculate quasisymmetric functions 𝔖 α * . We develop several formulas for the entries in the inverse of this matrix based on suitably generalized raising operators, tournaments, and special rim-hook tableaux. Our analysis reveals how the combinatorial conditions defining dual immaculate tableaux arise naturally from algebraic properties of raising operators. We also obtain an elementary combinatorial proof that the definition of 𝔖 α * via dual immaculate tableaux is equivalent to the definition of the immaculate noncommutative symmetric functions 𝔖 α via noncommutative Jacobi–Trudi determinants. A factorization of raising operators leads to bases of NSym interpolating between the 𝔖-basis and the h-basis, and bases of QSym interpolating between the 𝔖 * -basis and the M-basis. We also give t-analogues for most of these results using combinatorial statistics defined on dual immaculate tableaux and tournaments.

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DOI: https://doi.org/10.5802/alco.193
Classification: 05E05,  05A19
Keywords: Kostka matrix, quasisymmetric functions, noncommutative symmetric functions, dual immaculate tableaux, immaculate basis, special rim hook tableaux, tournaments
Loehr, Nicholas A. 1; Niese, Elizabeth 2

1. Virginia Tech Dept. of Mathematics Blacksburg, VA 24061, USA
2. Marshall University Dept. of Mathematics Huntington, WV 25755, USA
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Loehr, Nicholas A.; Niese, Elizabeth. Combinatorics of the immaculate inverse Kostka matrix. Algebraic Combinatorics, Volume 4 (2021) no. 6, pp. 1119-1142. doi : 10.5802/alco.193. https://alco.centre-mersenne.org/articles/10.5802/alco.193/

[1] Allen, Edward E.; Hallam, Joshua; Mason, Sarah K. Dual immaculate quasisymmetric functions expand positively into Young quasisymmetric Schur functions, J. Combin. Theory Ser. A, Volume 157 (2018), pp. 70-108 | Article | MR 3780408 | Zbl 1385.05072

[2] Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike A lift of the Schur and Hall–Littlewood bases to non-commutative symmetric functions, Canad. J. Math., Volume 66 (2014) no. 3, pp. 525-565 | Article | MR 3194160 | Zbl 1291.05206

[3] Blasiak, Jonah; Morse, Jennifer; Pun, Anna; Summers, Daniel Catalan functions and k-Schur positivity, J. Amer. Math. Soc., Volume 32 (2019) no. 4, pp. 921-963 | Article | MR 4013737 | Zbl 1423.05192

[4] Blasiak, Jonah; Morse, Jennifer; Pun, Anna; Summers, Daniel k-Schur expansions of Catalan functions, Adv. Math., Volume 371 (2020), Paper no. 107209, 39 pages | Article | MR 4118771 | Zbl 1443.05181

[5] Carbonara, Joaquin O. A combinatorial interpretation of the inverse t-Kostka matrix, Discrete Math., Volume 193 (1998) no. 1-3, pp. 117-145 Selected papers in honor of Adriano Garsia (Taormina, 1994) | Article | MR 1661366 | Zbl 1061.05504

[6] Eğecioğlu, Ömer; Remmel, Jeffrey B. A combinatorial interpretation of the inverse Kostka matrix, Linear and Multilinear Algebra, Volume 26 (1990) no. 1-2, pp. 59-84 | Article | MR 1034417 | Zbl 0735.05013

[7] Gelʼfand, Israel M.; Krob, Daniel; Lascoux, Alain; Leclerc, Bernard; Retakh, Vladimir S.; Thibon, Jean-Yves Noncommutative symmetric functions, Adv. Math., Volume 112 (1995) no. 2, pp. 218-348 | Article | MR 1327096 | Zbl 0831.05063

[8] Haglund, James; Luoto, Kurt; Mason, Sarah; van Willigenburg, Stephanie Quasisymmetric Schur functions, J. Combin. Theory Ser. A, Volume 118 (2011) no. 2, pp. 463-490 | Article | MR 2739497 | Zbl 1229.05270

[9] Hivert, Florent Hecke algebras, difference operators, and quasi-symmetric functions, Adv. Math., Volume 155 (2000) no. 2, pp. 181-238 | Article | MR 1794711 | Zbl 0990.05129

[10] Lascoux, Alain; Schützenberger, Marcel-Paul Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B, Volume 286 (1978) no. 7, pp. 323-324 | MR 472993 | Zbl 0374.20010

[11] Loehr, Nicholas A. Combinatorics, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2018, xxiv+618 pages | MR 3791447 | Zbl 1381.05001

[12] Loehr, Nicholas A.; Serrano, Luis G.; Warrington, Gregory S. Transition matrices for symmetric and quasisymmetric Hall–Littlewood polynomials, J. Combin. Theory Ser. A, Volume 120 (2013) no. 8, pp. 1996-2019 | Article | MR 3102172 | Zbl 1278.05241

[13] Luoto, Kurt; Mykytiuk, Stefan; van Willigenburg, Stephanie An introduction to quasisymmetric Schur functions: Hopf algebras, quasisymmetric functions, and Young composition tableaux, Springer Briefs in Mathematics, Springer, New York, 2013, xiv+89 pages | Article | MR 3097867 | Zbl 1277.16027

[14] Macdonald, Ian Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages | MR 1354144 | Zbl 0824.05059

[15] Sagan, Bruce E. The symmetric group: Representations, combinatorial algorithms, and symmetric functions, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | Article | MR 1824028 | Zbl 0964.05070

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