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A generalization of immanants based on partition algebra characters

Part of: Basic linear algebra Algebraic combinatorics

Published online by Cambridge University Press:  01 April 2024

John M. Campbell
John M. Campbell*
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, NS B3H 4R2, Canada
*
e-mail: jmaxwellcampbell@gmail.com
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Abstract

We introduce a generalization of immanants of matrices, using partition algebra characters in place of symmetric group characters. We prove that our immanant-like function on square matrices, which we refer to as the recombinant, agrees with the usual definition for immanants for the special case whereby the vacillating tableaux associated with the irreducible characters correspond, according to the Bratteli diagram for partition algebra representations, to the integer partition shapes for symmetric group characters. In contrast to previously studied variants and generalizations of immanants, as in Temperley–Lieb immanants and f-immanants, the sum that we use to define recombinants is indexed by a full set of partition diagrams, as opposed to permutations.

Keywords

Immanantpartition algebracharacterirreducible representation

MSC classification

Primary: 15A15: Determinants, permanents, other special matrix functions
Secondary: 05E10: Combinatorial aspects of representation theory

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 4 , December 2024 , pp. 1001 - 1010
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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References

Clearman, S., Shelton, B., and Skandera, M., Path tableaux and combinatorial interpretations of immanants for class functions on ${S}_n$ . In: 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Mathematics & Theoretical Computer Science, Nancy, 2011, pp. 233–244.Google Scholar
Coelho, M. P. and Duffner, M. A., Subspaces where an immanant is convertible into its conjugate . Linear Multilinear Algebra 48(2001), no. 4, 383–408.CrossRefGoogle Scholar
de Guise, H., Spivak, D., Kulp, J., and Dhand, I., $D$ -functions and immanants of unitary matrices and submatrices . J. Phys. A 49(2016), no. 9, 09LT01, 12 pp.CrossRefGoogle Scholar
Duffner, M. A., Guterman, A. E., and Spiridonov, I. A., Converting immanants on skew-symmetric matrices . Linear Algebra Appl. 618(2021), 76–96.CrossRefGoogle Scholar
Gamas, C., Spherical functions and immanants . Linear Multilinear Algebra 47(2000), no. 2, 151–173.CrossRefGoogle Scholar
Goulden, I. P. and Jackson, D. M., Immanants of combinatorial matrices . J. Algebra 148(1992), no. 2, 305–324.CrossRefGoogle Scholar
Greene, C., Proof of a conjecture on immanants of the Jacobi–Trudi matrix . Linear Algebra Appl. 171(1992), 65–79.CrossRefGoogle Scholar
Grone, R. and Merris, R., A Hadamard inequality for the second immanant . J. Algebra 111(1987), no. 2, 343–346.CrossRefGoogle Scholar
Haiman, M., Hecke algebra characters and immanant conjectures . J. Amer. Math. Soc. 6(1993), no. 3, 569–595.CrossRefGoogle Scholar
Halverson, T., Characters of the partition algebras . J. Algebra 238(2001), no. 2, 502–533.CrossRefGoogle Scholar
Halverson, T. and Ram, A., Partition algebras . European J. Combin. 26(2005), no. 6, 869–921.CrossRefGoogle Scholar
Hartmann, W., On the complexity of immanants . Linear Multilinear Algebra 18(1985), no. 2, 127–140.CrossRefGoogle Scholar
Heyfron, P., Some inequalities concerning immanants . Math. Proc. Cambridge Philos. Soc. 109(1991), no. 1, 15–30.CrossRefGoogle Scholar
Itoh, M., Twisted immanant and matrices with anticommuting entries . Linear Multilinear Algebra 64(2016), no. 8, 1637–1653.CrossRefGoogle Scholar
James, G., Hecke algebras and immanants . Linear Algebra Appl. 197/198(1994), 659–670.CrossRefGoogle Scholar
Jones, V. F. R., The Potts model and the symmetric group. In: Subfactors (Kyuzeso, 1993), World Scientific Publishing, River Edge, NJ, 1994, pp. 259–267.Google Scholar
Konvalinka, M., On Goulden–Jackson’s determinantal formula for the immanant . Ann. Comb. 13(2010), no. 4, 511–518.CrossRefGoogle Scholar
Kostant, B., Immanant inequalities and $0$ -weight spaces . J. Amer. Math. Soc. 8(1995), no. 1, 181–186.CrossRefGoogle Scholar
Littlewood, D. E. and Richardson, A. R., Group characters and algebra . Philos. Trans. Roy. Soc. Lond., Ser. A 233(1934), 99–142.Google Scholar
Marcott, C., Partition algebras and Kronecker coefficients. M.Math. Thesis, University of Waterloo, 2015.Google Scholar
Martin, P., Potts models and related problems in statistical mechanics, World Scientific Publishing, Teaneck, NJ, 1991.CrossRefGoogle Scholar
Martin, P., Temperley–Lieb algebras for nonplanar statistical mechanics—the partition algebra construction . J. Knot Theory Ramifications 3(1994), no. 1, 51–82.CrossRefGoogle Scholar
Martin, P., The structure of the partition algebras . J. Algebra 183(1996), no. 2, 319–358.CrossRefGoogle Scholar
Martin, P. P., The partition algebra and the Potts model transfer matrix spectrum in high dimensions . J. Phys. A 33(2000), no. 19, 3669–3695.CrossRefGoogle Scholar
Orellana, R. and Zabrocki, M., Symmetric group characters as symmetric functions . Adv. Math. 390(2021), Article no. 107943, 34 pp.CrossRefGoogle Scholar
Pate, T. H., Tensor inequalities, $\xi$ -functions and inequalities involving immanants . Linear Algebra Appl. 295(1999), no. 1-3, 31–59.CrossRefGoogle Scholar
Pylyavskyy, P., ${A}_2$ -web immanants . Discrete Math. 310(2010), nos. 15–16, 2183–2197.CrossRefGoogle Scholar
Rhoades, B. and Skandera, M., Temperley–Lieb immanants . Ann. Comb. 9(2005), no. 4, 451–494.CrossRefGoogle Scholar
Rhoades, B. and Skandera, M., Kazhdan–Lusztig immanants and products of matrix minors . J. Algebra 304(2006), no. 2, 793–811.CrossRefGoogle Scholar
Rhoades, B. and Skandera, M., Kazhdan–Lusztig immanants and products of matrix minors. II . Linear Multilinear Algebra 58(2010), nos. 1–2, 137–150.CrossRefGoogle Scholar
Stanley, R. P. and Stembridge, J. R., On immanants of Jacobi–Trudi matrices and permutations with restricted position . J. Combin. Theory Ser. A 62(1993), no. 2, 261–279.CrossRefGoogle Scholar
Tabata, R., Limiting behavior of immanants of certain correlation matrix . Linear Algebra Appl. 510(2016), 230–245.CrossRefGoogle Scholar