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Higher spin representations of the Yangian of $\mathfrak{sl}_2$ and R-matrices

Yaping Yang, Paul Zinn-Justin

TL;DR

This work geometrizes higher spin representations of the Yangian Y_{\hbar}(\mathfrak{sl}_2) using critical cohomology of a framed quiver with potential, encoding spin data through a w-fold framing and a rank-1 reduction to a 2d setting. It builds a lattice-model interpretation whose partition function matches weight functions produced by a framed shuffle algebra, and constructs an explicit action of Y_{\hbar}(\mathfrak{sl}_2) on the torus-fixed cohomology, recovering the evaluation representations for w=1 and revealing a fusion-tensor structure for general w. The R-matrices are computed from fixed-point data, with transparent results for w=1 and w=2, and a general w=2 formula enabling explicit checks of Yang–Baxter relations. Finally, a coherent picture is drawn linking weight functions, lattice models, framed shuffle, and stable envelopes, providing a geometric route to Kirillov–Reshetikhin-type representations and their intertwiners.

Abstract

We study higher spin (pure and mixed spin) representations of the Yangian of $\mathfrak{sl}_2$. We provide a geometric realization in terms of the critical cohomology of representations of the quiver with potential of Bykov and Zinn-Justin [BZJ20]. When the framing dimension is 1, it recovers the evaluation pullback of the $\ell+1$-dimensional irreducible representation of $\mathfrak{sl}_2$. We introduce the lattice model and prove that its partition function coincides with the weight function constructed using the framed shuffle formula. The latter follows the approach of Rimányi, Tarasov and Varchenko [RTV15].

Higher spin representations of the Yangian of $\mathfrak{sl}_2$ and R-matrices

TL;DR

This work geometrizes higher spin representations of the Yangian Y_{\hbar}(\mathfrak{sl}_2) using critical cohomology of a framed quiver with potential, encoding spin data through a w-fold framing and a rank-1 reduction to a 2d setting. It builds a lattice-model interpretation whose partition function matches weight functions produced by a framed shuffle algebra, and constructs an explicit action of Y_{\hbar}(\mathfrak{sl}_2) on the torus-fixed cohomology, recovering the evaluation representations for w=1 and revealing a fusion-tensor structure for general w. The R-matrices are computed from fixed-point data, with transparent results for w=1 and w=2, and a general w=2 formula enabling explicit checks of Yang–Baxter relations. Finally, a coherent picture is drawn linking weight functions, lattice models, framed shuffle, and stable envelopes, providing a geometric route to Kirillov–Reshetikhin-type representations and their intertwiners.

Abstract

We study higher spin (pure and mixed spin) representations of the Yangian of . We provide a geometric realization in terms of the critical cohomology of representations of the quiver with potential of Bykov and Zinn-Justin [BZJ20]. When the framing dimension is 1, it recovers the evaluation pullback of the -dimensional irreducible representation of . We introduce the lattice model and prove that its partition function coincides with the weight function constructed using the framed shuffle formula. The latter follows the approach of Rimányi, Tarasov and Varchenko [RTV15].
Paper Structure (32 sections, 26 theorems, 254 equations, 1 figure)

This paper contains 32 sections, 26 theorems, 254 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lattice model formulation
  3. Dimension reductions and torus fixed points
  4. The dimension reduction
  5. Torus fixed points
  6. The case when $w=1$
  7. Proof of Theorem \ref{['thm:action']} and Proposition \ref{['prop:w=1']}
  8. Proof of Theorem \ref{['thm:action']}
  9. The Yangian of $\mathfrak{sl}_2$
  10. The action of the Yangian
  11. Proof of Proposition \ref{['prop:ef formula']}
  12. The proof of Proposition \ref{['prop:w=1']}
  13. Geometric interpretation
  14. Torus fixed points on $\tilde{X}(v, \vec{1}_w)^{st}/G_v$
  15. Framed shuffle formula
  16. ...and 17 more sections

Key Result

Theorem A

For any $w\in {\mathbb N}$ and any $(\ell_1, \cdots, \ell_w)\in {\mathbb N}^w$, the Yangian $Y_{\hbar}(\mathfrak{sl}_2)$ acts on the cohomology

Figures (1)

  • Figure 1: An example of state of a lattice model with $v=4$, $w=6$, and $(v_1,\ldots,v_w)=(0,1,0,2,0,1)$.

Theorems & Definitions (54)

  • Theorem A
  • Proposition 1.1
  • Theorem B
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 44 more