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Fermionic Characters and Arbitrary Highest-Weight Integrable -Modules

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Abstract

This paper contains the generalization of the Feigin-Stoyanovsky construction to all integrable -modules. We give formulas for the q-characters of any highest-weight integrable module of as a linear combination of the fermionic q-characters of the fusion products of a special set of integrable modules. The coefficients in the sum are the entries of the inverse matrix of generalized Kostka polynomials in q −1. We prove the conjecture of Feigin and Loktev regarding the q-multiplicities of irreducible modules in the graded tensor product of rectangular highest weight-modules in the case of . We also give the fermionic formulas for the q-characters of the (non-level-restricted) fusion products of rectangular highest-weight integrable -modules.

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References

  1. Ardonne, E., Bouwknegt, P., Schoutens, K.: Non-abelian quantum Hall states–-exclusion statistics, K-matrices, and duality. J. Stat. Phys. 102(3–4), 421–469 (2001)

  2. Ardonne, E., Kedem, R., Stone, M.: Filling the bose sea: symmetric quantum Hall edge states and affine characters. J. Phys. A 38(3), 617–636 (2005)

    Article  MathSciNet  Google Scholar 

  3. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241(2), 333–380 (1984)

    Article  MathSciNet  Google Scholar 

  4. Feigin, B., Jimbo, M., Kedem, R., Loktev, S., Miwa, T.: Spaces of coinvariants and fusion product, affine sl 2 character formulas in terms of kostka polynomials. Duke Math. J. 125(3), 549–588 (2004)

    Article  MathSciNet  Google Scholar 

  5. Feigin, B., Jimbo, M., Loktev, S., Miwa, T., Mukhin, E.: Addendum to: ``Bosonic formulas for (k,l)-admissible partitions''. Ramanujan J. 7(4), 519–530 (2003)

    Article  MathSciNet  Google Scholar 

  6. Feigin, B., Kedem, R., Loktev, S., Miwa, T., Mukhin, E.: Combinatorics of the spaces of coinvariants. Transform. Groups 6(1), 25–52 (2001)

    Article  MathSciNet  Google Scholar 

  7. Feigin, B., Kedem, R., Loktev, S., Miwa, T., Mukhin, E.: Combinatorics of the coinvariants: dual functional realization and recursion. Compositio Math. 134(2), 193–241 (2002)

    Article  MathSciNet  Google Scholar 

  8. Feigin, B., Loktev, S.: On generalized Kostka polynomials and the quantum Verlinde rule. In: Differential topology, infinite-dimensional Lie algebras, and applications, Volume 194 of Amer. Math. Soc. Transl. Ser. 2, Providence, RI: Amer. Math. Soc., 1999, pp. 61–79

  9. Georgiev, G.: Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space. http://arxiv.org/list/math.QA/9504024, 1995

  10. Georgiev, G.: Combinatorial constructions of modules for infinite-dimensional Lie algebras, I. Principal subspace. J. Pure Appl. Alg. 112(3), 247–286 (1996)

    Article  MathSciNet  Google Scholar 

  11. Kac, V.G.: Infinite-dimensional Lie algebras. Cambridge: Cambridge University Press, Third edition, 1990

  12. Kedem, R., Klassen, T.R., McCoy, B.M., Melzer, E.: Fermionic quasi-particle representations for characters of (G (1))1× (G (1))1/(G (1))2. Phys. Lett. B 304(3–4), 263–270 (1993)

    Google Scholar 

  13. Kedem, R., McCoy, B.M.: Construction of modular branching functions from Bethe's equations in the 3-state Potts chain. J. Stat. Phys. 71(5–6), 865–901 (1993)

    Google Scholar 

  14. Kirillov, A.N.: Ubiquity of Kostka polynomials. In: Physics and combinatorics 1999 (Nagoya), River Edge, NJ: World Sci. Publishing, 2001, pp. 85–200

  15. Kirillov, A.N., Schilling, A., Shimozono, M.: A bijection between Littlewood-Richardson tableaux and rigged configurations. Selecta Math. (N.S.) 8(1), 67–135 (2002)

    Article  MathSciNet  Google Scholar 

  16. Kirillov, A.N., Shimozono, M.: A generalization of the Kostka-Foulkes polynomials. J. Alg. Combin. 15(1), 27–69 (2002)

    Article  Google Scholar 

  17. Lepowsky, J., Primc, M.: Structure of the standard modules for the affine Lie algebra A [1] 1. Volume 46 of Contemporary Mathematics. Providence, RI: Amer. Math. Soc., 1985

  18. Macdonald, I.G.: Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. New York: Clarendon Press Oxford University Press, Second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications

  19. Moore, G., Read, N.: Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360(2–3), 362–396 (1991)

    Google Scholar 

  20. Primc, M.: Vertex operator construction of standard modules for A (1) n . Pac. J. Math. 162(1), 143–187 (1994)

    MathSciNet  Google Scholar 

  21. Primc, M.: Loop modules in annihilating ideals of standard modules for affine Lie algebras. In: VII. Mathematikertreffen Zagreb-Graz (Graz, 1990), Volume 313 of Grazer Math. Ber., Karl-Franzens-Univ. Graz, 1991, pp. 39–44

  22. Schilling, A., Ole Warnaar, S.: Inhomogeneous lattice paths, generalized Kostka polynomials and A n-1 supernomials. Commun. Math. Phys. 202(2), 359–401 (1999)

    Article  MathSciNet  Google Scholar 

  23. Stoyanovskii, A.V., Feigin, B.L.: Functional models of the representations of current algebras, and semi-infinite Schubert cells. Funkt. Anal. i Pril. 28(1), 68–90, 96 (1994)

    MathSciNet  Google Scholar 

  24. Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory on P 1 and monodromy representations of braid group. In: Conformal field theory and solvable lattice models (Kyoto, 1986), Volume 16 of Adv. Stud. Pure Math., Boston, MA: Academic Press, 1988, pp. 297–372

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Authors and Affiliations

  1. Department of Physics, University of Illinois, 1110 W. Green St., Urbana, IL, 61801, USA

    Eddy Ardonne & Michael Stone

  2. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    Rinat Kedem

Authors
  1. Eddy Ardonne
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  2. Rinat Kedem
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  3. Michael Stone
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Correspondence to Eddy Ardonne.

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Communicated by L. Takhtajan

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Ardonne, E., Kedem, R. & Stone, M. Fermionic Characters and Arbitrary Highest-Weight Integrable -Modules. Commun. Math. Phys. 264, 427–464 (2006). https://doi.org/10.1007/s00220-005-1486-3

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  • DOI: https://doi.org/10.1007/s00220-005-1486-3

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