Geodesic
Weight $q$-multiplicities for representations of $\mathfrak{sp}_4(\mathbb{C})$
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 10 (2017) no. 4, pp. 494-502.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we present a closed formula for the values of the $q$-analog of Kostant's partition function for the Lie algebra $\mathfrak{sp}_4(\mathbb{C})$ and use this result to give a simple formula for the $q$-multiplicity of a weight in the representations of the Lie algebra $\mathfrak{sp}_4(\mathbb{C})$. This generalizes the 2012 work of Refaghat and Shahryari that presented a closed formula for weight multiplicities in representations of the Lie algebra $\mathfrak{sp}_4(\mathbb{C})$.
Keywords: Sympletic Lie algebra, $q$-analog of Kostant partition function, weight multiplicity, weight $q$-multiplicity.
Mots-clés : Kostant partition functions 
@article{JSFU_2017_10_4_a9,
     author = {Pamela E. Harris and Edward L. Lauber},
     title = {Weight $q$-multiplicities for representations of $\mathfrak{sp}_4(\mathbb{C})$},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {494--502},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2017_10_4_a9/}
}
TY  - JOUR
AU  - Pamela E. Harris
AU  - Edward L. Lauber
TI  - Weight $q$-multiplicities for representations of $\mathfrak{sp}_4(\mathbb{C})$
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2017
SP  - 494
EP  - 502
VL  - 10
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2017_10_4_a9/
LA  - en
ID  - JSFU_2017_10_4_a9
ER  - 
%0 Journal Article
%A Pamela E. Harris
%A Edward L. Lauber
%T Weight $q$-multiplicities for representations of $\mathfrak{sp}_4(\mathbb{C})$
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2017
%P 494-502
%V 10
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2017_10_4_a9/
%G en
%F JSFU_2017_10_4_a9
Pamela E. Harris; Edward L. Lauber. Weight $q$-multiplicities for representations of $\mathfrak{sp}_4(\mathbb{C})$. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 10 (2017) no. 4, pp. 494-502. http://geodesic.mathdoc.fr/item/JSFU_2017_10_4_a9/

[1] J. Fernández-Nú{ñ}ez, W. García-Fuertes, A. M. Perelomov, “On the generating function of weight multiplicities for the representations of the Lie algebra $C_2$”, Journal of Mathematical Physics, 56:4 (2015), 041702 | DOI | MR | Zbl

[2] R. Goodman, N. R. Wallach, Symmetry, Representations and Invariants, Springer, New York, 2009 | MR | Zbl

[3] P. E. Harris, Computing weight multiplicities, Foundations for Undergraduate Research in Mathematics, Springer (to appear)

[4] P. E. Harris, “On the adjoint representation of $\mathfrak{sl}_n$ and the Fibonacci numbers”, Comptes Rendus de l Académie des Sciences, Series I, Paris, 349 (2011), 935–937 | MR | Zbl

[5] P. E. Harris, Kostant's weight multiplicity formula and the Fibonacci numbers, arXiv: 1111.6648 [math.RT]

[6] P. E. Harris, Combinatorial Problems Related to Kostant's Weight Multiplicity Formula, Doctoral dissertation, University of Wisconsin-Milwaukee, Milwaukee, WI, 2012 | MR | Zbl

[7] P. E. Harris, E. Insko, M. Omar, The $q$-analog of Kostant's partition function and the highest root of the classical Lie algebras, arXiv: 1508.07934 [math.CO]

[8] P. E. Harris, E. Insko, L. K. Williams, “The adjoint representation of a classical Lie algebra and the support of Kostant's weight multiplicity formula”, Journal of Combinatorics, 7:1 (2016), 75–116 | DOI | MR | Zbl

[9] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Universty Press, Cambridge, United Kingdom, 1997 | MR

[10] B. Kostant, “A formula for the multiplicity of a weight”, Proc. Natl. Acad. Sci, USA, 44 (1958), 588–589 | DOI | MR | Zbl

[11] G. Lusztig, “Singularities, character formulas, and a $q$-analog of weight multiplicities”, Astérisque, 101–102, 1983, 208–229 | MR | Zbl

[12] H. Refaghat, M. Shahryari, “Kostant Partition Function for $\frak{sp}_4(\mathbb{C})$”, Journal of Siberian Federal University. Mathematics Physics, 5:1 (2012), 18–24