×

zbMATH — the first resource for mathematics

The \(q\)-analog of Kostant’s partition function and the highest root of the simple Lie algebras. (English) Zbl 1459.17021
Summary: For a given weight of a complex simple Lie algebra, the \(q\)-analog of Kostant’s partition function is a polynomial valued function in the variable \(q\), where the coefficient of \(q^k\) is the number of ways the weight can be written as a nonnegative integral sum of exactly \(k\) positive roots. In this paper we determine generating functions for the \(q\)-analog of Kostant’s partition function when the weight in question is the highest root of the classical Lie algebras of types \(B\), \(C\), and \(D\), and the exceptional Lie algebras of type \(G_2\), \(F_4\), \(E_6\), \(E_7\), and \(E_8\).

MSC:
17B20 Simple, semisimple, reductive (super)algebras
17B25 Exceptional (super)algebras
05A17 Combinatorial aspects of partitions of integers
PDF BibTeX XML Cite
Full Text: Link
References:
[1] W. Baldoni, M. Beck, C. Cochet and M. Vergne, Volume computation for polytopes and partition functions for classical root systems, Discrete Comput. Geom. 35 (2006), 551–595. · Zbl 1105.52001
[2] W. Baldoni and M. Vergne, Kostant partitions functions and flow polytopes, Transform. Groups 13 (2008), 447–469. · Zbl 1200.52008
[3] A. Barvinok, A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res. 19 (1994), 769–779. · Zbl 0821.90085
[4] A. Barvinok, Lattice points and lattice polytopes, in Handbook of discrete and computational geometry, CRC Press Ser. Discrete Math. Appl., CRC, Boca Raton, FL, 1997, 133–152. · Zbl 0912.52009
[5] A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, in New perspectives in algebraic combinatorics, (Berkeley, CA, 1996–97), Math. Sci. Res. Inst. Publ. Vol. 38, Cambridge Univ. Press, Cambridge, 1999, 91–147. · Zbl 0940.05004
[6] A. D. Berenstein and A. V. Zelevinsky, Tensor product multiplicities and convex polytopes in partition space, J. Geom. Phys. 5 (1988), 453–472.
[7] S. Billey, V. Guillemin and E. Rassart, A vector partition function for the multiplicities ofslkC, J. Algebra 278 (2004), 251–293. · Zbl 1116.17005
[8] S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, Ann. Comb. 13 (2010), 413–424. · Zbl 1231.05009
[9] C. Cochet, Vector partition function and representation theory, Conf. Proc. Formal Power Series and Algebraic Combin. (2005), p. 12. · Zbl 1078.22006
[10] R. W. Deckhart, On the combinatorics of Kostant’s partition function, J. Algebra 96 (1985), 9–17. · Zbl 0603.17005
[11] J. Fernández Núñez, W. García Fuertes and A. M. Perelomov, Generating functions and multiplicity formulas: the case of rank two simple lie algebras,arXiv:1506.07815(2015).
[12] R. Goodman and N. R. Wallach, Symmetry, representations, and invariants, Graduate Texts in Math. Vol. 255, Springer, Dordrecht, 2009. · Zbl 1173.22001
[13] R. K. Gupta, Characters and the q-analog of weight multiplicity, J. London Math. Soc. 36 (2) (1987), 68–76. · Zbl 0649.17009
[14] P. E. Harris, Kostant’s weight multiplicity formula and the Fibonacci numbers,arXiv:1111.6648(2011). P.E. HARRIS ET AL. / AUSTRALAS. J. COMBIN. 71 (1) (2018), 68–9191
[15] P. E. Harris, On the adjoint representation ofslnand the Fibonacci numbers, C. R. Math. Acad. Sci. Paris 349 (2011), 935–937. · Zbl 1273.17010
[16] P. E. Harris, Combinatorial problems related to Kostant’s weight multiplicity formula, ProQuest LLC, Ann Arbor, MI, 2012. Ph.D. Thesis, The University of Wisconsin—Milwaukee.
[17] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Math. Vol. 9, Springer-Verlag, New York-Berlin, 1978; Second printing, revised. · Zbl 0447.17001
[18] B. Kostant, A formula for the multiplicity of a weight, Trans. Amer. Math. Soc. 93 (1959), 53–73. · Zbl 0131.27201
[19] G. Lusztig, Singularities, character formulas, and a q-analog of weight multiplicities, in Analysis and topology on singular spaces II, III (Luminy, 1981), Astérisque Vol. 101, Soc. Math. France, Paris, 1983, 208–229.
[20] K. Mészáros and A. H. Morales, Flow polytopes of signed graphs and the Kostant partition function, Int. Math. Res. Not. IMRN (2015), 830–871. · Zbl 1307.05097
[21] J. R. Schmidt and A. M. Bincer, The Kostant partition function for simple Lie algebras, J. Math. Phys. 25 (1984), 2367–2373. · Zbl 0551.17002
[22] T. Tate and S. Zelditch, Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers, J. Funct. Anal. 217 (2004), 402–447. · Zbl 1062.22026
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.