Colored Bosonic Models and Matrix Coefficients

Abstract

We develop the theory of colored bosonic models (initiated by Borodin and Wheeler). We will show how a family of such models can be used to represent the values of Iwahori vectors in the "spherical model" of representations of $GL_r(F)$, where $F$ is a nonarchimedean local field. Among our results are a monochrome factorization, which is the realization of the Boltzmann weights by fusion of simpler weights, a local lifting property relating the colored models with uncolored models, and an action of the affine Hecke algebra on the partition functions of a particular family of models by Demazure-Lusztig operators. As an application of the local lifting property we reprove a theorem of Korff evaluating the partition functions of the uncolored models in terms of Hall-Littlewood plynomials. Our results are very closely parallel to the theory of fermionic models representing Iwahori Whittaker functions developed by Brubaker, Buciumas, Bump and Gustafsson, with many striking relationships between the two theories, confirming the philosophy that the spherical and Whittaker models of principal series representations are dual.

Figures (13)

• Figure 1: The grid with boundary conditions for the uncolored model, $\mathfrak{S}^P_\lambda(\mathbf{z};t)$ or $\mathfrak{S}^R_\lambda(\mathbf{z}; t)$, corresponding to the partition $\lambda=(8,6,6,1,0)$, with $r=5$. A state of the model will assign spins to the interior edges.
• Figure 2: Uncolored Boltzmann weights of two types: the $P$-weights (which coincide with those in KorffVerlinde, and the $R$-weights.
• Figure 3: The Yang-Baxter equation. The partition functions of the two small 3-vertex systems are the same. Here $a,b,c,d,e,f$ are the fixed boundary spins, and in each case we sum over the possible spins of the three interior edges. We may use either the $P$- or the $R$-weights. Uncolored case: Use the weights from Figure \ref{['fig:uncolored_weights']} and the R-matrix from Figure \ref{['fig:uncolored_rmatrix']}. Colored case: Use the colored weights obtained by fusion (Figure \ref{['fig:fusion']}) from the monochrome weights (Figure \ref{['fig:monochrome']}).
• Figure 4: The uncolored R-matrix. This works for either the uncolored $R$- or $P$-models.
• Figure 5: Monochrome vertex type $z_i,c$. The vertical edge can carry only the color $c$. The possible states of the horizontal edges are all colors and $+$. Possible spins of the vertical edges are labeled by integers $n$, representing $n$ copies of the boson of color $c$.

Theorems & Definitions (39)

• Remark 1.1
• Theorem 1.2: BBBGIwahori, Theorem A, NaprienkoWhittaker, Section 3.3.1
• Remark 1.3
• Remark 1.4
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3: Demazure
• proof