Browse issues
no. 1
Hook-shape immanant characters from Stanley–Stembridge characters
Lesnevich, Nathan R. T.1
1 Washington University in St Louis Department of Mathematics One Brookings Drive St Louis, MO 63130
Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 137-157.

We consider the Schur-positivity of monomial immanants of Jacobi–Trudi matrices, in particular whether a non-negative coefficient of the trivial Schur function implies non-negative coefficients for other Schur functions in said immanants. We prove that this true for hook-shape Schur functions using combinatorial methods in a representation theory setting. Our main theorem proves that hook-shape immanant characters can be written as finite non-negative integer sums of Stanley–Stembridge characters, and provides an explicit combinatorial formula for these sums. This resolves a special case of a longstanding conjecture of Stanley and Stembridge that posits such a sum exists for all immanant characters. We also provide several simplifications for computing immanant characters, and several corollaries applying the main result to cases where the coefficient of the trivial Schur function in monomial immanants of Jacobi–Trudi matrices is known to be non-negative.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.331
Classification: 05E05, 05E10
Keywords: immanants, Jacobi–Trudi matrices

Lesnevich, Nathan R. T. 1

1 Washington University in St Louis Department of Mathematics One Brookings Drive St Louis, MO 63130
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2024__7_1_137_0,
     author = {Lesnevich, Nathan R. T.},
     title = {Hook-shape immanant characters from {Stanley{\textendash}Stembridge} characters},
     journal = {Algebraic Combinatorics},
     pages = {137--157},
     publisher = {The Combinatorics Consortium},
     volume = {7},
     number = {1},
     year = {2024},
     doi = {10.5802/alco.331},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.331/}
}
TY  - JOUR
AU  - Lesnevich, Nathan R. T.
TI  - Hook-shape immanant characters from Stanley–Stembridge characters
JO  - Algebraic Combinatorics
PY  - 2024
SP  - 137
EP  - 157
VL  - 7
IS  - 1
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.331/
DO  - 10.5802/alco.331
LA  - en
ID  - ALCO_2024__7_1_137_0
ER  - 
%0 Journal Article
%A Lesnevich, Nathan R. T.
%T Hook-shape immanant characters from Stanley–Stembridge characters
%J Algebraic Combinatorics
%D 2024
%P 137-157
%V 7
%N 1
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.331/
%R 10.5802/alco.331
%G en
%F ALCO_2024__7_1_137_0
Lesnevich, Nathan R. T. Hook-shape immanant characters from Stanley–Stembridge characters. Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 137-157. doi : 10.5802/alco.331. https://alco.centre-mersenne.org/articles/10.5802/alco.331/

[1] Abreu, Alex; Nigro, Antonio Chromatic symmetric functions from the modular law, J. Combin. Theory Ser. A, Volume 180 (2021), Paper no. 105407, 30 pages | DOI | MR | Zbl

[2] Brosnan, Patrick; Chow, Timothy Y. Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties, Adv. Math., Volume 329 (2018), pp. 955-1001 | DOI | MR | Zbl

[3] Dahlberg, Samantha Triangular Ladders P d,2 are e-positive, 2019 | arXiv

[4] Gasharov, Vesselin Incomparability graphs of (3+1)-free posets are s-positive, Discrete Math., Volume 157 (1996) no. 1-3, pp. 193-197 | DOI | MR | Zbl

[5] Goulden, I. P.; Jackson, D. M. Immanants of combinatorial matrices, J. Algebra, Volume 148 (1992) no. 2, pp. 305-324 | DOI | MR | Zbl

[6] Greene, Curtis Proof of a conjecture on immanants of the Jacobi–Trudi matrix, Linear Algebra Appl., Volume 171 (1992), pp. 65-79 | DOI | MR | Zbl

[7] Guay-Paquet, Mathieu A modular relation for the chromatic symmetric functions of (3+1)-free posets, 2013 | arXiv

[8] Haiman, Mark Hecke algebra characters and immanant conjectures, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 569-595 | DOI | MR | Zbl

[9] Harada, Megumi; Precup, Martha E. The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture, Algebr. Comb., Volume 2 (2019) no. 6, pp. 1059-1108 | DOI | Numdam | MR | Zbl

[10] 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

[11] Shareshian, John; Wachs, Michelle L. Chromatic quasisymmetric functions, Adv. Math., Volume 295 (2016), pp. 497-551 | DOI | MR | Zbl

[12] Stanley, Richard P. A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math., Volume 111 (1995) no. 1, pp. 166-194 | DOI | MR | Zbl

[13] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR

[14] Stanley, Richard P. Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR

[15] Stanley, Richard P.; Stembridge, John R. On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A, Volume 62 (1993) no. 2, pp. 261-279 | DOI | MR | Zbl

[16] Stembridge, John R. Some conjectures for immanants, Canad. J. Math., Volume 44 (1992) no. 5, pp. 1079-1099 | DOI | MR | Zbl

Cited by Sources: