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Twisted vertex representations via spin groups and the McKay correspondence

Igor Frenkel, Naihuan Jing, Weiqiang Wang

Abstract

We establish a twisted analog of our recent work on vertex representations and the McKay correspondence. For each finite group $Γ$ and a virtual character of $Γ$ we construct twisted vertex operators on the Fock space spanned by the super spin characters of the spin wreath products $Γ\wr\widetilde{S}_n$ of $Γ$ and a double cover of the symmetric group $S_n$ for all $n$. When $Γ$ is a subgroup of $SL_2(\mathbb C)$ with the McKay virtual character, our construction gives a group theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras. When $Γ$ is an arbitrary finite group and the virtual character is trivial, our vertex operator construction yields the spin character tables for $Γ\wr\widetilde{S}_n$.

Twisted vertex representations via spin groups and the McKay correspondence

Abstract

We establish a twisted analog of our recent work on vertex representations and the McKay correspondence. For each finite group and a virtual character of we construct twisted vertex operators on the Fock space spanned by the super spin characters of the spin wreath products of and a double cover of the symmetric group for all . When is a subgroup of with the McKay virtual character, our construction gives a group theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras. When is an arbitrary finite group and the virtual character is trivial, our vertex operator construction yields the spin character tables for .

Paper Structure

This paper contains 36 sections, 39 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. A double cover of the wreath product
  3. The spin group $\widetilde{S}_{n}$
  4. The spin wreath product $\widetilde{\Gamma}_{n}$
  5. Conjugacy classes of $\widetilde{\Gamma}_{n}$
  6. Split conjugacy classes of $\widetilde{\Gamma}_{n}$
  7. The Hopf algebra $R^-_{\Gamma}$ of super spin characters
  8. Superalgebras and supermodules
  9. Induced supermodules
  10. The space $R^-(\widetilde{\Gamma}_{n})$
  11. Hopf algebra structure on $R^-_{\Gamma}$
  12. Basic spin representations of $\widetilde{\Gamma}_{n}$
  13. A weighted bilinear form on $R(\Gamma)$ and $R^-(\widetilde{\Gamma}_{n})$
  14. Basic spin representations
  15. The spin character $\pi_n(\gamma)$ of $\widetilde{\Gamma}_{n}$
  16. ...and 21 more sections

Key Result

Proposition 2.1

Jo Each element of $\widetilde{S}_{n}$ can be presented as where $\{i_1\cdots i_m\}, \{j_1\cdots j_k\}, \cdots$ is a partition of the set $\{1, 2, \cdots, n\}$ and $p=0, 1$. If $z^pc_1c_2\cdots c_l=z^{p'}c'_1 c'_2\cdots c'_{l'}$ are two expressions of the same element in terms of cycles $c_i$ and $c_i'$, then $l=l'$ and there is a permutation $\sigma\in S_l where $|c_i|$ denotes the length of the

Theorems & Definitions (64)

  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Theorem 2.5
  • proof
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • ...and 54 more