### Abstract

In this paper, we continue our study of the Hilbert polynomials of coinvariants begun in our previous work [B. Feigin et al., math.QA/0205324, 2002]. We describe the sl_{n} fusion products for symmetric tensor representations following the method of [B. Feigin, E. Feigin, math.QA/0201111, 2002], and show that their Hilbert polynomials are A_{n-1}-supernomials. We identify the fusion product of arbitrary irreducible sl_{n} -modules with the fusion product of their restriction to sl_{n-1}. Then using the equivalence theorem from [B. Feigin et al., math.QA/0205324, 2002] and the results above for sl_{3} we give a fermionic formula for the Hilbert polynomials of a class of sl_{2} coinvariants in terms of the level-restricted Kostka polynomials. The coinvariants under consideration are a generalization of the coinvariants studied in [B. Feigin et al., Transfom. Groups 6 (2001) 25-52; math.QA/0009198, 2000; math.QA/0012190, 2000]. Our formula differs from the fermionic formula established in [B. Feigin et al., Transfom. Groups 6 (2001) 25-52; QA/0012190 math.QA/0012190, 2000] and implies the alternating sum formula conjectured in [B. Feigin, S.; Loktev, QA/9812093 1998; Amer. Math. Sci. Transl. 194 (1999) 61-79] for this case.

Original language | English (US) |
---|---|

Pages (from-to) | 147-179 |

Number of pages | 33 |

Journal | Journal of Algebra |

Volume | 279 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1 2004 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

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_{2}character formulas in terms of Kostka polynomials.

*Journal of Algebra*,

*279*(1), 147-179. https://doi.org/10.1016/j.jalgebra.2004.03.004

**Spaces of coinvariants and fusion product II. sl _{2} character formulas in terms of Kostka polynomials.** / Feigin, B.; Jimbo, M.; Kedem, R.; Loktev, S.; Miwa, T.

Research output: Contribution to journal › Article

_{2}character formulas in terms of Kostka polynomials',

*Journal of Algebra*, vol. 279, no. 1, pp. 147-179. https://doi.org/10.1016/j.jalgebra.2004.03.004

_{2}character formulas in terms of Kostka polynomials. Journal of Algebra. 2004 Sep 1;279(1):147-179. https://doi.org/10.1016/j.jalgebra.2004.03.004

}

TY - JOUR

T1 - Spaces of coinvariants and fusion product II. sl2 character formulas in terms of Kostka polynomials

AU - Feigin, B.

AU - Jimbo, M.

AU - Kedem, R.

AU - Loktev, S.

AU - Miwa, T.

PY - 2004/9/1

Y1 - 2004/9/1

N2 - In this paper, we continue our study of the Hilbert polynomials of coinvariants begun in our previous work [B. Feigin et al., math.QA/0205324, 2002]. We describe the sln fusion products for symmetric tensor representations following the method of [B. Feigin, E. Feigin, math.QA/0201111, 2002], and show that their Hilbert polynomials are An-1-supernomials. We identify the fusion product of arbitrary irreducible sln -modules with the fusion product of their restriction to sln-1. Then using the equivalence theorem from [B. Feigin et al., math.QA/0205324, 2002] and the results above for sl3 we give a fermionic formula for the Hilbert polynomials of a class of sl2 coinvariants in terms of the level-restricted Kostka polynomials. The coinvariants under consideration are a generalization of the coinvariants studied in [B. Feigin et al., Transfom. Groups 6 (2001) 25-52; math.QA/0009198, 2000; math.QA/0012190, 2000]. Our formula differs from the fermionic formula established in [B. Feigin et al., Transfom. Groups 6 (2001) 25-52; QA/0012190 math.QA/0012190, 2000] and implies the alternating sum formula conjectured in [B. Feigin, S.; Loktev, QA/9812093 1998; Amer. Math. Sci. Transl. 194 (1999) 61-79] for this case.

AB - In this paper, we continue our study of the Hilbert polynomials of coinvariants begun in our previous work [B. Feigin et al., math.QA/0205324, 2002]. We describe the sln fusion products for symmetric tensor representations following the method of [B. Feigin, E. Feigin, math.QA/0201111, 2002], and show that their Hilbert polynomials are An-1-supernomials. We identify the fusion product of arbitrary irreducible sln -modules with the fusion product of their restriction to sln-1. Then using the equivalence theorem from [B. Feigin et al., math.QA/0205324, 2002] and the results above for sl3 we give a fermionic formula for the Hilbert polynomials of a class of sl2 coinvariants in terms of the level-restricted Kostka polynomials. The coinvariants under consideration are a generalization of the coinvariants studied in [B. Feigin et al., Transfom. Groups 6 (2001) 25-52; math.QA/0009198, 2000; math.QA/0012190, 2000]. Our formula differs from the fermionic formula established in [B. Feigin et al., Transfom. Groups 6 (2001) 25-52; QA/0012190 math.QA/0012190, 2000] and implies the alternating sum formula conjectured in [B. Feigin, S.; Loktev, QA/9812093 1998; Amer. Math. Sci. Transl. 194 (1999) 61-79] for this case.

UR - http://www.scopus.com/inward/record.url?scp=4043154411&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4043154411&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2004.03.004

DO - 10.1016/j.jalgebra.2004.03.004

M3 - Article

AN - SCOPUS:4043154411

VL - 279

SP - 147

EP - 179

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -