Duke Mathematical Journal

Spaces of coinvariants and fusion product, I: From equivalence theorem to Kostka polynomials

B. Feigin, M. Jimbo, R. Kedem, S. Loktev, and T. Miwa

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Abstract

The fusion rule gives the dimensions of spaces of conformal blocks in Wess-Zumino-Witten (WZW) theory. We prove a dimension formula similar to the fusion rule for spaces of coinvariants of affine Lie algebras \widehat{\mathfrak{g}}. An equivalence of filtered spaces is established between spaces of coinvariants of two objects: highest weight \widehat{\mathfrak{g}}-modules and tensor products of finite-dimensional evaluation representations of \mathfrak{g}\otimes \mathbb{C}[t].

In the \widehat{\mathfrak{sl}}2-case we prove that their associated graded spaces are isomorphic to the spaces of coinvariants of fusion products and that their Hilbert polynomials are the level-restricted Kostka polynomials.

Article information

Source
Duke Math. J. Volume 125, Number 3 (2004), 549-588.

Dates
First available in Project Euclid: 18 November 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1100793679

Digital Object Identifier
doi:10.1215/S0012-7094-04-12533-3

Mathematical Reviews number (MathSciNet)
MR2166753

Zentralblatt MATH identifier
02141302

Subjects
Primary: 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65]

Citation

Feigin, B.; Jimbo, M.; Kedem, R.; Loktev, S.; Miwa, T. Spaces of coinvariants and fusion product, I: From equivalence theorem to Kostka polynomials. Duke Math. J. 125 (2004), no. 3, 549--588. doi:10.1215/S0012-7094-04-12533-3. http://projecteuclid.org/euclid.dmj/1100793679.


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References

  • L. Begin, A. N. Kirillov, P. Mathieu, and M. A. Walton, Berenstein-Zelevinski triangles, elementary couplings, and fusion rules, Lett. Math. Phys. 28 (1993), 257--268.
  • B. L. Feigin and E. Feigin, $Q$-characters of the tensor products in $\mathfraksl_2$-case, Mosc. Math. J. 2 (2002), 567--588.
  • B. L. Feigin and D. B. Fuchs, ``Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody algebras'' in Geometry and Physics: Essays in Honor of I. M. Gel'fand, reprint of J. Geom Phys. 5, North Holland, Amsterdam, 1991, 209--235.
  • B. Feigin, R. Kedem, S. Loktev, T. Miwa, and E. Mukhin, Combinatorics of the $\widehat\mathfraksl_2$ spaces of coinvariants, Transform. Groups 6 (2001), 25--52.
  • --. --. --. --., Combinatorics of the $\widehat\mathfraksl_2$ coinvariants: Dual functional realization and recursion, Compositio Math. 134 (2002), 193--241.
  • --. --. --. --., Combinatorics of the $\widehat\mathfraksl_2$ spaces of coinvariants: Loop Heisenberg modules and recursion, Selecta Math. (N.S.) 8 (2002), 419--474.
  • B. L. Feigin and S. Loktev, ``On generalized Kostka polynomials and quantum Verlinde rule'' in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2 194, Amer. Math. Soc., Providence, 1999, 61--79.
  • H. O. Foulkes, ``A survey of some combinatorial aspects of symmetric functions'' in Permutations (Paris, 1972), Gauthier-Villars, Paris, 1974.
  • W. Fulton and R. MacPherson, A compactfication of configuration spaces, Ann. of Math. (2) 139 (1994), 183--225.
  • G. Hatayama, A. N. Kirillov, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Character formulae of $\widehat\sll_n$-modules and inhomogeneous paths, Nuclear Phys. B 536 (1999), 575--616.
  • R. Kedem, T. R. Klassen, B. M. McCoy, and E. Melzer, Fermionic sum representations for conformal field theory characters, Phys. Lett. B 307 (1993), 68--76.
  • R. Kedem and B. M. McCoy, Construction of modular branching functions from Bethe's equations in the $3$-state Potts chain, J. Statist. Phys. 71 (1993), 875--901.
  • A. N. Kirillov and N. Yu. Reshetikhin, The Bethe Ansatz and the combinatorics of Young tableaux, J. Soviet Math. 41 (1988), 925--955.
  • A. N. Kirillov, A. Schilling, and M. Shimozono, A bijection between Littlewood-Richardson tableaux and rigged configurations, Selecta Math. (N.S.) 8 (2002), 67--135.
  • A. N. Kirillov and M. Shimozono, A generalization of the Kostka-Foulkes polynomials, J. Algebraic Combin. 15 (2002), 27--69.
  • A. Lascoux and M.-P. Schützenberger, Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Ser. A-B 286 (1978), A323--A324.
  • I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.
  • A. Nakayashiki and Y. Yamada, Kostka polynomials and energy functions in solvable lattice models, Selecta Math. (N.S.) 3 (1997), 547--599.
  • A. Schilling and M. Shimozono, Fermionic formulas for level-restricted generalized Kostka polynomials and coset branching functions, Comm. Math. Phys. 220 (2001), 105--164.
  • A. Schilling and O. Warnaar, Supernomial coefficients, polynomial identities and $q$-series, Ramanujan J. 2 (1998), 459--494.
  • --. --. --. --., Inhomogeneous lattice paths, generalized Kostka polynomials and $A_n-1$ supernomials, Comm. Math. Phys. 202 (1999), 359--401.
  • A. V. Stoyanovsky and B. L. Feigin, Functional models for representations of current algebras and semi-infinite Schubert cells, Funct. Anal. Appl. 28 (1993), 55--72.
  • A. Tsuchiya, K. Ueno, and Y. Yamada, ``Conformal field theory on universal family of stable curves with gauge symmetries'' in Integrable Systems in Quantum Field Theory and Statistical Mechanics, Adv. Stud. Pure Math. 19, Academic Press, Boston, 1989, 459--566.