×

zbMATH — the first resource for mathematics

Computing weight \(q\)-multiplicities for the representations of the simple Lie algebras. (English) Zbl 1436.17011
Summary: The multiplicity of a weight \(\mu \) in an irreducible representation of a simple Lie algebra \(\mathfrak {g}\) with highest weight \(\lambda \) can be computed via the use of Kostant’s weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a \(q\)-analog of Kostant’s weight multiplicity and present a SageMath program to compute \(q\)-multiplicities for the simple Lie algebras.

MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
17-08 Computational methods for problems pertaining to nonassociative rings and algebras
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Baldoni, W; Beck, M; Cochet, C; Vergne, M, Volume computation for polytopes and partition functions for classical root systems, Discret. Comput. Geom., 35, 551-595, (2006) · Zbl 1105.52001
[2] Baldoni, W; Vergne, M, Kostant partitions functions and flow polytopes, Transform. Groups, 13, 447-469, (2008) · Zbl 1200.52008
[3] Barvinok, A, A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res., 19, 769-779, (1994) · Zbl 0821.90085
[4] Barvinok, A.: Lattice points and lattice polytopes. In: Handbook of Discrete and Computational Geometry, CRC Press Ser. Discrete Math. Appl., pp. 133-152. CRC, Boca Raton (1997). , · Zbl 0912.52009
[5] Barvinok, A., Pommersheim, J.E.: An algorithmic theory of lattice points in polyhedra. In New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-97). Volume 38 of Math. Sci. Res. Inst. Publ., pp. 91-147. Cambridge University Press, Cambridge (1999) · Zbl 0940.05004
[6] Berenstein, AD; Zelevinsky, AV, Tensor product multiplicities and convex polytopes in partition space, J. Geom. Phys., 5, 453-472, (1988) · Zbl 0712.17006
[7] Billey, S; Guillemin, V; Rassart, E, A vector partition function for the multiplicities of \(\mathfrak{sl}_k(\mathbb{C})\), J. Algebra, 278, 251-293, (2004) · Zbl 1116.17005
[8] Cochet, C.: Vector partition function and representation theory. In: Conference Proceedings Formal Power Series and Algebraic Combinatorics, p. 12, 2005 · Zbl 1078.22006
[9] Deckart, RW, On the combinatorics of kostant’s partition function, J. Algebra, 96, 9-17, (1985) · Zbl 0603.17005
[10] Fernández-Núñez, J; García-Fuertes, W; Perelomov, AM, On the generating function of weight multiplicities for the representations of the Lie algebra \(C_2\), J. Math. Phys., 56, 041702, (2015) · Zbl 1387.17018
[11] Goodman, R., Wallach, N.R.: Symmetry. Representations and Invariants. Springer, New York (2009) · Zbl 1173.22001
[12] Gupta, RK, Characters and the \(q\)-analog of weight multiplicity, J. Lond. Math. Soc., 2, 68-76, (1987) · Zbl 0649.17009
[13] Harris, P.E.: Chapter 9. In: Wootton, A., Peterson, V., Lee, C. (eds.) A Primer for Undergraduate Research, Foundations for Undergraduate Research in Mathematics. Birkhäuser, Basel (to appear)
[14] Harris, P.E.: Combinatorial problems related to Kostant’s weight multiplicity formula. Doctoral dissertation, University of Wisconsin-Milwaukee, Milwaukee, WI (2012)
[15] Harris, P.E.: Kostant’s weight multiplicity formula and the Fibonacci numbers. arXiv:1111.6648 [math.RT] · Zbl 0649.17009
[16] Harris, PE, On the adjoint representation of \(\mathfrak{sl}_n\) and the Fibonacci numbers, C. R. Math. Acad. Sci. Paris, 349, 935-937, (2011) · Zbl 1273.17010
[17] Harris, P.E., Insko, E., Omar, M.: The \(q\)-analog of Kostant’s partition function and the highest root of the simple Lie algebras (2016). http://arxiv.org/pdf/1508.07934 · Zbl 1387.17018
[18] Harris, P.E., Insko, E., Simpson, A.: Computing weight \(q\)-multiplicities for the representations of the simple Lie algebras (2017). http://arxiv.org/pdf/1710.02183 · Zbl 1436.17011
[19] Harris, P.E., Insko, E., Simpson, A.: GitHub code download. https://github.com/antman1935/lie_algebras
[20] Harris, P; Insko, E; Williams, L, The adjoint representation of a classical Lie algebra and the support of kostant’s weight multiplicity formula, J. Comb., 7, 75-116, (2016) · Zbl 1330.05164
[21] Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Universty Press, Cambridge (1997) · Zbl 0725.20028
[22] Kostant, B, A formula for the multiplicity of a weight, Proc. Nat. Acad. Sci. USA, 44, 588-589, (1958) · Zbl 0081.02202
[23] Knapp, A.W.: Lie Groups Beyond an Introduction. Birkhäuser Boston Inc., Boston (2002) · Zbl 1075.22501
[24] Lusztig, G, Singularities, character formulas, and a \(q\)-analog of weight multiplicities, Astérisque, 101, 208-229, (1983) · Zbl 0561.22013
[25] Sagan, B.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, pp. 141-144. Springer, New York (2001) · Zbl 0964.05070
[26] Schmidt, JR; Bincer, AM, The Kostant partition function for simple Lie algebras, J. Math. Phys., 25, 2367-2373, (1984) · Zbl 0551.17002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.