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.

Received:
Revised:
Accepted:
Published online:
DOI: 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2021__4_6_1119_0,
     author = {Loehr, Nicholas A. and Niese, Elizabeth},
     title = {Combinatorics of the immaculate inverse {Kostka} matrix},
     journal = {Algebraic Combinatorics},
     pages = {1119--1142},
     publisher = {MathOA foundation},
     volume = {4},
     number = {6},
     year = {2021},
     doi = {10.5802/alco.193},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.193/}
}
TY  - JOUR
AU  - Loehr, Nicholas A.
AU  - Niese, Elizabeth
TI  - Combinatorics of the immaculate inverse Kostka matrix
JO  - Algebraic Combinatorics
PY  - 2021
SP  - 1119
EP  - 1142
VL  - 4
IS  - 6
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.193/
DO  - 10.5802/alco.193
LA  - en
ID  - ALCO_2021__4_6_1119_0
ER  - 
%0 Journal Article
%A Loehr, Nicholas A.
%A Niese, Elizabeth
%T Combinatorics of the immaculate inverse Kostka matrix
%J Algebraic Combinatorics
%D 2021
%P 1119-1142
%V 4
%N 6
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.193/
%R 10.5802/alco.193
%G en
%F ALCO_2021__4_6_1119_0
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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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) | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

Cited by Sources: