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Higgsing $qq$-character and irreducibility

Taro Kimura

Abstract

We study the $qq$-character of quantum affine and toroidal algebra modules, with a focus on the role of spectral parameters. In particular, we revisit how their specialization affects the irreducibility of these modules.

Higgsing $qq$-character and irreducibility

Abstract

We study the -character of quantum affine and toroidal algebra modules, with a focus on the role of spectral parameters. In particular, we revisit how their specialization affects the irreducibility of these modules.
Paper Structure (34 sections, 23 theorems, 161 equations, 6 figures)

This paper contains 34 sections, 23 theorems, 161 equations, 6 figures.

Table of Contents

  1. Introduction and summary
  2. Finite-dimensional module of quantum affine algebra and $qq$-character
  3. Irreducibility of $qq$-character
  4. Higgs mechanism
  5. Summary and organization
  6. Construction of $qq$-character
  7. Quiver
  8. $\mathscr{S}$-function
  9. Vertex operators
  10. iWeyl reflection
  11. Contour integral formula
  12. $A_1$ quiver
  13. Weight two
  14. Weight three
  15. General weight
  16. ...and 19 more sections

Key Result

Theorem 1.1

The image of the $q$-character equals the intersection of the kernels of the screening operators.

Figures (6)

  • Figure 1: Hasse diagram for the iWeyl reflection flow of the weight $w=(2,0)$ for $A_2$ quiver.
  • Figure 2: Hasse diagram for the iWeyl reflection flow of the weight $w=(0,2)$ for $A_2$ quiver.
  • Figure 3: Hasse diagram for the iWeyl reflection flow of the weight $w= (1,1)$ for $A_2$ quiver.
  • Figure 4: Hasse diagram for the iWeyl reflection flow of the weight $w= (2,0)$ for $B_2/C_2$ quiver
  • Figure 5: Hasse diagram for the iWeyl reflection flow of the weight $w=(0,2)$ for $B_2/C_2$ quiver.
  • ...and 1 more figures

Theorems & Definitions (50)

  • Theorem 1.1: Frenkel:1999ky; Conjecture in Frenkel:1998ojj
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Conjecture 1.6
  • Remark 1.7
  • Conjecture 1.8: Theorems \ref{['thm:A_1_I']}, \ref{['thm:A_1_II']} for $A_1$ quiver
  • Definition 2.1
  • Remark 2.2
  • ...and 40 more