2.1 Demazure characters and atoms

Given $s_i$ , a simple reflection, we can define the associated isobaric Demazure operator acting on $f \in \mathcal {O} (T)$ as

(2.1) $$ \begin{align} \pi_i f (\mathbf{z}) = \frac{f(\mathbf{z}) - \mathbf{z}^{-\alpha_i} f (s_i \mathbf{z})}{1 - \mathbf{z}^{-\alpha_i}}. \end{align} $$

The numerator is divisible by the denominator, so the resulting function is again in $\mathcal {O}(T)$ .

One can check that $\pi ^2_i = \pi _i = s_i \pi _i$ . Given any $\mu \in \Lambda $ , set $k = \langle \mu , \alpha _i^{\vee } \rangle $ so $s_i (\mu ) = \mu - k\alpha _i$ . Then the action on the monomial $\mathbf {z}^{\mu }$ is given explicitly by

(2.2) $$ \begin{align} \pi_i \mathbf{z}^{\mu} = \begin{cases} \mathbf{z}^{\mu} +\mathbf{z}^{\mu - \alpha_i} + \cdots +\mathbf{z}^{s_i (\mu)} & \text{if } k \geqslant 0, \\ 0 & \text{if } k = - 1,\\ - (\mathbf{z}^{\mu + \alpha_i} +\mathbf{z}^{\mu + 2 \alpha_i} + \cdots + \mathbf{z}^{s_i (\mu + \alpha_i)}) & \text{if } k < - 1. \end{cases} \end{align} $$

Define $\overline {\pi }_i := \pi _i - 1$ . Explicitly, we have

(2.3) $$ \begin{align} \overline{\pi}_i f(\mathbf{z}) := \frac{f(\mathbf{z}) - f(s_i\mathbf{z})}{\mathbf{z}^{\alpha_i} - 1}. \end{align} $$

Both $\pi _i$ and $\overline {\pi }_i$ satisfy the braid relations. Thus, for any $w \in W$ , we can choose any reduced word $w = s_{i_1} \dotsm s_{i_k}$ to define $\pi _w = \pi _{i_1} \cdots \pi _{i_k}$ and $\overline {\pi }_w = \overline {\pi }_{i_1} \cdots \overline {\pi }_{i_k}$ by Matsumoto’s theorem [Reference Matsumoto60]. For $w = 1$ , we set $\pi _1 = \overline {\pi }_1 = 1$ .

For $\lambda $ , a dominant weight, let $\chi _{\lambda }$ denote the character of the irreducible representation $V_{\lambda }$ with highest-weight $\lambda $ . The ${\color {darkred}{\textit {Demazure character formula}}}$ is the identity, for $\mathbf {z}\in T$ ,

$$\begin{align*}\chi_{\lambda} (\mathbf{z}) = \pi_{w_0} \mathbf{z}^{\lambda}. \end{align*}$$

For a proof, see [Reference Bump21, Thm. 25.3]. More generally, for any Weyl group element w, we may consider $\pi _w\mathbf {z}^{\lambda }$ and $\overline {\pi }_w\mathbf {z}^{\lambda }$ . These polynomials are called ${\color {darkred}{\textit {Demazure characters}}}$ and ${\color {darkred}{\textit {Demazure atoms}}}$ , respectively. The following relation between the two is well-known.

Theorem 2.1 [Reference Lascoux and Schützenberger55]; see also [Reference Pun67, Lemma 2.5].

Let $f\in \mathcal {O}(T)$ . Then

(2.4) $$ \begin{align} \pi_{w} f(\mathbf{z}) = \sum_{y \leqslant w} \overline{\pi}_{y} f(\mathbf{z}). \end{align} $$

As a corollary, we obtain the following decomposition of characters in terms of Demazure atoms.

Corollary 2.2. We have

$$\begin{align*}\operatorname{ch} V_{\lambda} = \pi_{w_0} \mathbf{z}^{\lambda} = \sum_{y} \overline{\pi}_{y} \mathbf{z}^{\lambda}. \end{align*}$$

2.2 Signed permutations and the Weyl group action

For the remainder of the paper, we will only consider Cartan types $BC$ . We identify the maximal torus T with the space $(\mathbb {C}^*)^n$ , where n is the rank of the group. The Weyl group W of type $B_n$ is isomorphic to the Weyl of type $C_n$ , and it is known as the hyperoctohedral group. It is generated by the simple reflections $s_i$ for $i \in \{1, \dotsc , n\}$ subject to the relations

(2.5) $$ \begin{align} \begin{aligned} s_{n}s_{n-1}s_ns_{n-1} &= s_{n-1} s_ns_{n-1}s_n, \\ s_i s_{i+1} s_i &= s_{i+1} s_i s_{i+1}, & &\mbox{ if } i < n-1\\ s_i s_j &= s_j s_i, && \mbox{ if } |i-j| \geq 2\\ s_i^2 &= 1. \end{aligned} \end{align} $$

The Weyl group acts on elements $\mathbf {z} \in (\mathbb {C}^*)^n$ as follows:

(2.6a) $$ \begin{align} s_i (\ldots, z_i, z_{i+1}, \ldots) & = (\ldots, z_{i+1}, z_{i},\ldots), && \text{if } i < n, \end{align} $$

(2.6b) $$ \begin{align} s_n (\ldots, z_{n-1}, z_{n}) & = (\ldots, z_{n-1}, z_{n}^{-1}), && \text{if } i = n. \end{align} $$

The elements of W can be explicitly described using ${\color {darkred}{\textit {signed permutations}}}$ of n, which are permutations of

$$\begin{align*}1 < 2 < \cdots < n < \overline{n} < \cdots < \overline{1} \end{align*}$$

such that $w(i) = \overline {w(\overline {\imath })}$ . Here we use the convention that $\overline {\overline {\imath }}=i$ . Thus we can determine a signed permutation by the image of $1 \leq i \leq n$ . The simple transposition for $i < n$ is given by $s_i = (i \; i+1)$ , and $s_n$ sends $n \leftrightarrow \overline {n}$ . An ${\color {darkred}{\textit {inversion}}}$ is a pair $1 \leq i < j \leq n$ such that $w(i)> w(j)$ . The longest element $w_0$ is given by the signed permutation $[\overline {1}, \overline {2}, \dotsc , \overline {n}]$ .

The subgroup of W generated by $s_i$ for $i<n$ is a subgroup isomorphic to the Weyl group of type A. We shall denote the subgroup by $W^A$ .

2.3 Functional equations

We now discuss the explicit functional equations for the Demazure characters and atoms that will be used to prove the main theorems in this paper.

We first consider Demazure characters. equation (2.1) can be written explicitly as

(2.7a) $$ \begin{align} \pi_i f(\mathbf{z}) & := \dfrac{f(\mathbf{z}) - z_i^{-1} z_{i+1} f(s_i\mathbf{z})}{1 - z_i^{-1} z_{i+1}}, && \text{if } i<n, \end{align} $$

(2.7b) $$ \begin{align} \pi_n f(\mathbf{z}) & := \dfrac{f(\mathbf{z}) - z_n^{-1}f(s_n\mathbf{z})}{1 - z_n^{-1}}, && \text{in type B,} \end{align} $$

(2.7c) $$ \begin{align} \pi_n f(\mathbf{z}) & := \dfrac{f(\mathbf{z}) - z_n^{-2}f(s_n\mathbf{z})}{1 - z_n^{-2}}, && \text{in type C.} \end{align} $$

Let us denote $D_w(\mathbf {z},\lambda ) := \pi _w z^{\lambda }$ . Let $s_i$ be a simple reflection and $w \in W$ such that $\ell (s_i w)>\ell (w)$ . From equation (2.7), we deduce the following:

(2.8a) $$ \begin{align} (z_i -z_{i+1})D_{s_iw}(\mathbf{z}, \lambda) & = z_{i} D_{w}(\mathbf{z}, \lambda) - z_{i+1} D_{w}(s_i\mathbf{z}, \lambda), && \text{if } i < n, \end{align} $$

(2.8b) $$ \begin{align} (z_n -1)D_{s_n w}(\mathbf{z}, \lambda) & = z_n D_{w}(\mathbf{z}, \lambda) - D_{w}(s_n\mathbf{z}, \lambda), && \text{in type B,} \end{align} $$

(2.8c) $$ \begin{align} (z_{n}^2 -1)D_{s_n w}(\mathbf{z}, \lambda) & = z_n^2 D_{w}(\mathbf{z}, \lambda) - D_{w}(s_n\mathbf{z}, \lambda), && \text{in type C.} \end{align} $$

Next, we consider the Demazure atoms. In this case, equation (2.3) can be rewritten as

(2.9a) $$ \begin{align} \overline{\pi}_i f(\mathbf{z}) & := \dfrac{f(\mathbf{z})-f(s_i\mathbf{z})}{z_i z_{i+1}^{-1}-1}, && \text{if } i<n, \end{align} $$

(2.9b) $$ \begin{align} \overline{\pi}_n f(\mathbf{z}) & := \dfrac{f(\mathbf{z})-f(s_n\mathbf{z})}{z_n-1}, && \text{in type B,} \end{align} $$

(2.9c) $$ \begin{align} \overline{\pi}_n f(\mathbf{z}) & := \dfrac{f(\mathbf{z})-f(s_n\mathbf{z})}{z_n^2-1}, && \text{in type C.} \end{align} $$

Let us denote $A_w(\mathbf {z},\lambda ) := \overline {\pi }_w z^{\lambda }$ . Let $s_i$ be a simple reflection and $w \in W$ such that $\ell (s_i w)>\ell (w)$ . We rewrite the equation above as

(2.10a) $$ \begin{align} (z_i -z_{i+1})A_{s_iw}(\mathbf{z}, \lambda) & = z_{i+1} \bigl( A_{w}(\mathbf{z}, \lambda) - A_{w}(s_i\mathbf{z}, \lambda) \bigr), && \text{if } i < n, \end{align} $$

(2.10b) $$ \begin{align} (z_n -1)A_{s_n w}(\mathbf{z}, \lambda) & = A_{w}(\mathbf{z}, \lambda) - A_{w}(s_n\mathbf{z}, \lambda), && \text{in type B,} \end{align} $$

(2.10c) $$ \begin{align} (z_{n}^2 -1)A_{s_n w}(\mathbf{z}, \lambda) & = A_{w}(\mathbf{z}, \lambda) - A_{w}(s_n\mathbf{z}, \lambda), && \text{in type C.} \end{align} $$

3.1 Billiards

Our first goal is to prove the following ‘type A’ functional equation for the partition function.

Lemma 3.7. Choose $i < n$ , $j=i+1$ and $w \in W$ such that $\ell (s_i w) = \ell (w) + 1$ . Then we have

(3.3) $$ \begin{align} (z_i - z_j) Z(\overline{\mathfrak{S}}_{\lambda, s_i w}; \mathbf{z}) = z_j \bigl( Z(\overline{\mathfrak{S}}_{\lambda, w}; \mathbf{z}) - z_i z_j^{-1} Z(\overline{\mathfrak{S}}_{\lambda, w}; s_i \mathbf{z}) \bigr). \end{align} $$

Proof. We prove the functional equation by following the sequence of steps pictured in Figure 9). In each step, we exhibit a model, and models in two consecutive steps will have the same partition function, possibly up to some factor. Finally, by comparing the partition functions of the first and last models, we prove the result.

1. 1. In the first step of Figure 9, we add the ‘double R-matrix’ to the left of the model $\overline {\mathfrak {S}}_{\lambda , s_iw}$ . With the imposed left boundary condition, the double R-matrix must be in one of the following two admissible configurations:

   
   Here the colours satisfy ${\color {red} c}> {\color {blue} c'}$ because $\ell (s_i w) = \ell (w) + 1$ . We conclude that the partition function of the model in the first step is
   $$\begin{align*}z_i (z_i^{-1} - z_j^{-1}) z_j z_j Z(\overline{\mathfrak{S}}_{\lambda, s_i w}; \mathbf{z}) + z_i z_i^{-1} z_j z_j Z(\overline{\mathfrak{S}}_{\lambda, w}; \mathbf{z}). \end{align*}$$

2. 2. To obtain the second model, we pass the three R-matrices to the right by using the Yang–Baxter equation in Proposition 3.3, leaving behind the $R^{\Gamma }_{\Delta }$ -matrix on the left.

3. 3. We apply the reflection equation (Proposition 3.4) to the $R^{\Delta }_{\Delta }$ -matrix and $R^{\Delta }_{\Gamma }$ -matrix. This contributes a factor of $\alpha =z_i^{-2}$ .

4. 4. We apply the unitary equation for the square of the $R^{\Gamma }_{\Gamma }$ -matrix. This contributes a factor of $\beta = z_i z_j$ to the partition function (Proposition 3.5).

5. 5. We use the Yang–Baxter equation to pass back the remaining $R^{\Delta }_{\Gamma }$ -matrix.

6. 6. We use a weak version of the unitary equation (in the sense that we only do it for certain fixed boundary values) for the $R^{\Delta }_{\Gamma }$ -matrix and $R^{\Gamma }_{\Delta }$ -matrix with a boundary condition ${\color {blue} c'}$ on the top left and top right and $0$ on the bottom left and bottom right. In terms of equation (3.2), we consider $a = {\color {blue} c'}$ and $b = 0$ . We only need to consider this boundary equation here because of the boundary conditions for $\mathfrak {S}_{\lambda , s_i w}$ . This contributes a factor of $\gamma = z_i^2$ times the partition function of $Z(\mathfrak {S}_{\lambda , s_i w}; s_i \mathbf {z})$ .

Comparing the initial and final models up to the contributions highlighted above, we obtain the equations

$$\begin{align*}z_i (z_i^{-1} - z_j^{-1}) z_j z_j Z(\overline{\mathfrak{S}}_{\lambda, s_i w}; \mathbf{z}) + z_i z_i^{-1} z_j z_j Z(\overline{\mathfrak{S}}_{\lambda, w}; \mathbf{z}) = \alpha \beta \gamma Z(\overline{\mathfrak{S}}_{\lambda, s_i w}; s_i \mathbf{z}). \end{align*}$$

Using the fact that $\alpha =z_i^{-2}, \beta = z_i z_j$ , and $\gamma = z_i^2$ and some basic algebra we can manipulate the equation above to obtain the desired result in equation (3.3).

Figure 9 Pictorial description of the sequence of steps to compute the action of the ith atom operator. Note that all the $\Gamma $ rows are coloured in blue and all the $\Delta $ rows are coloured in red. The R-matrices are determined by the colour of the row. For example, the leftmost R-matrix in the first step is the $R^{\Gamma }_{\Delta }\ R$ -matrix.

3.2 Ichthyology

We now study a version of the fish equation that we use to show a functional equation corresponding to the last simple reflection $s_n$ . We do not actually prove the usual fish equation but instead dissect it to its component pieces to obtain the desired functional equation.

Lemma 3.8. Choose $w \in W$ such that $\ell (s_n w) = \ell (w) + 1$ . Then we have

$$ \begin{align*} (z_n^2 - 1) Z(\overline{\mathfrak{S}}_{\lambda, s_n w}; \mathbf{z}) & = Z(\overline{\mathfrak{S}}_{\lambda, w}; \mathbf{z}) - Z(\overline{\mathfrak{S}}_{\lambda, w}; s_n \mathbf{z}) && (\text{type C}), \\ (z_n - 1) Z(\overline{\mathfrak{S}}_{\lambda, s_n w}; \mathbf{z}) & = Z(\overline{\mathfrak{S}}_{\lambda, w}; \mathbf{z}) - Z(\overline{\mathfrak{S}}_{\lambda, w}; s_n \mathbf{z}) && (\text{type B}). \end{align*} $$

Proof. The proof is the same for both cases outside of one computation where the K-matrix appears (recall that the L-matrix weights are the same for type B and type C). We proceed with one proof and are precise where the difference between types occur. Subsequently, we denote the Boltzmann weight of $\mathtt {k}_1$ by K.

In this proof, we work with models consisting of two rows connected by a U-turn on the right. These can be thought of as the last two rows in our previous model; therefore, we can use the fact that at most one unbarred and one barred colour will appear in this model (and if both do, they will be a pair: i.e., the barred colour will be the bar of the unbarred colour). We first follow the idea in [Reference Ivanov37] to modify the model from a $^{\Gamma }_{\Delta }$ model to a $^{\Gamma }_{\Gamma }$ model. To do this, for each admissible state in the row, we interchange the colour and noncolour in the last row $d \leftrightarrow 0$ . Thus we are now using the $\Gamma $ -weights in Figure 1 for the last row. We will now show that the partition function does not change under this process.

Note that the weight of every entry of the $\Delta \ L$ -matrix where the bottom value is $0$ equals the weight of the corresponding $\Gamma \ L$ -matrix with the interchanging $d \leftrightarrow 0$ . From the boundary conditions, this induces a bijection on the states of the model and preserves the Boltzmann weight contributions from the L-matrices. Therefore, the bottom two rows of the model are interchanged as follows:

where we have drawn the $^{\Gamma }_{\Delta }$ model on the left and $^{\Gamma }_{\Gamma }$ model on the right, with u being an unbarred colour such that $c \in \{u, \overline {u}\}$ .

The K-matrix on the right will be called the $K^{\Gamma }_{\Gamma }$ -matrix, and its Boltzmann weights are presented in Figure 10. For our purposes, we only require $H_1$ , $\overline {H}_1$ , $H_2$ and $\overline {H}_2$ to be any nonzero complex numbers; otherwise there will be no restrictions.

Figure 10 The coloured $K^{\Gamma }_{\Gamma }$ -matrix weights with ${\color {UQgold} u}$ being any unbarred colour.

Let $\widetilde {\mathfrak {S}}_{\mathtt {v}}$ denote the states of the $^{\Gamma }_{\Gamma }$ model for w with a fixed entry $\mathtt {v}$ from the $K^{\Gamma }_{\Gamma }$ -matrix. Then we have

$$ \begin{align*} Z(\overline{\mathfrak{S}}_{\lambda, w}; \mathbf{z}) & = K \overline{H}_1^{-1} Z(\widetilde{\mathfrak{S}}_{\overline{\mathtt{h}}_1}; \mathbf{z}) + H_2^{-1} Z(\widetilde{\mathfrak{S}}_{\mathtt{h}_2}; \mathbf{z}) + \overline{H}_2^{-1} Z(\widetilde{\mathfrak{S}}_{\overline{\mathtt{h}}_2}; \mathbf{z}). \end{align*} $$

Next attach on the left a $R^{\Gamma }_{\Gamma }$ -matrix, and then apply the standard train argument to pass the $R^{\Gamma }_{\Gamma }$ -matrix to the right side, as in Figure 11. Let $\widetilde {\mathfrak {S}}^d_{d'}$ denote the $^{\Gamma }_{\Gamma }$ model with the K-matrix removed; the bottom two right boundary conditions are d above and $d'$ below (either of which could be $0$ ). Therefore, from the $R^{\Gamma }_{\Gamma }$ -matrix and the Yang–Baxter equation, we have

$$ \begin{align*} z_n^{-1} \overline{H}_1^{-1} Z(\widetilde{\mathfrak{S}}_{\overline{\mathtt{h}}_1}; \mathbf{z}) & = z_n Z(\widetilde{\mathfrak{S}}_{\overline{u}}^0; s_n \mathbf{z}), \\ z_n^{-1} H_2^{-1} Z(\widetilde{\mathfrak{S}}_{\mathtt{h}_2}; \mathbf{z}) & = (z_n - z_n^{-1}) Z(\widetilde{\mathfrak{S}}_u^0; s_n \mathbf{z}) + z_n^{-1} Z(\widetilde{\mathfrak{S}}_0^u; s_n \mathbf{z}), \\ z_n^{-1} \overline{H}_2^{-1} Z(\widetilde{\mathfrak{S}}_{\overline{\mathtt{h}}_2}; \mathbf{z}) & = (z_n - z_n^{-1}) Z(\widetilde{\mathfrak{S}}_{\overline{u}}^0; s_n \mathbf{z}) + z_n^{-1} Z(\widetilde{\mathfrak{S}}_0^{\overline{u}}; s_n \mathbf{z}), \end{align*} $$

where u is an unbarred colour. In particular, to obtain the right-hand sides, we have the following possible ${\color {darkred}{\textit {fish}}}$ , local configurations of an $R^{\Gamma }_{\Gamma }$ -matrix and $K^{\Gamma }_{\Gamma }$ -matrix:

Therefore, we have

$$ \begin{align*} Z(\overline{\mathfrak{S}}_{\lambda, w}; \mathbf{z}) & = K z_n^2 Z(\widetilde{\mathfrak{S}}_{\overline{u}}^0; s_n \mathbf{z}) + z_n\bigl( (z_n - z_n^{-1}) Z(\widetilde{\mathfrak{S}}_u^0; s_n \mathbf{z}) + z_n^{-1} Z(\widetilde{\mathfrak{S}}_0^u; s_n \mathbf{z}) \bigr) \\ & \hspace{20pt} + z_n \bigl( (z_n - z_n^{-1}) Z(\widetilde{\mathfrak{S}}_{\overline{u}}^0; s_n \mathbf{z}) + z_n^{-1} Z(\widetilde{\mathfrak{S}}_0^{\overline{u}}; s_n \mathbf{z}) \bigr) \\ & = (K z_n^2 + z_n^2 - 1) \overline{K}^{-1} Z(\overline{\mathfrak{S}}_0^0(w); s_n \mathbf{z}) + Z(\overline{\mathfrak{S}}_{\overline{u}}^u(w); s_n \mathbf{z}) + Z(\overline{\mathfrak{S}}_{\overline{u}}^{\overline{u}}(w); s_n \mathbf{z}) \\ & \hspace{20pt} + (z_n^2 - 1) \overline{K}^{-1} Z(\overline{\mathfrak{S}}_0^0(s_n w); s_n \mathbf{z}), \end{align*} $$

where $\overline {\mathfrak {S}}_{d'}^d(w)$ denotes the model $\overline {\mathfrak {S}}_{\lambda , w}$ with a fixed K-matrix $^{d}_{d'}$ on the bottom two rows, and $\overline {K}$ denotes the parameter K but with $z_n \mapsto z_n^{-1}$ .

Figure 11 Left: The model $\widetilde {\mathfrak {S}}_{\mathtt {h}_2}$ with an R-matrix attached on the left. Right: The model after using the Yang–Baxter equation and the standard train argument.

Note that $Z(\overline {\mathfrak {S}}_0^0(s_n w); s_n \mathbf {z}) = Z(\overline {\mathfrak {S}}_{\lambda , s_n w}; s_n \mathbf {z})$ . To obtain our desired functional equations, we need to group the first three terms together into $Z(\overline {\mathfrak {S}}_{\lambda , w}; s_n \mathbf {z})$ . Hence, we require

(3.4) $$ \begin{align} (K z_n^2 + z_n^2 - 1) \overline{K}^{-1} = 1. \end{align} $$

The K-matrix for type C, where $K = z_n^{-2}$ , satisfies equation (3.4). Therefore, we have

$$ \begin{align*} Z(\overline{\mathfrak{S}}^C_{\lambda, w}; \mathbf{z}) & = Z(\overline{\mathfrak{S}}^C_{\lambda, w}; s_n \mathbf{z}) + (1- z_n^{-2}) Z(\overline{\mathfrak{S}}^C_{\lambda, s_n w}; s_n \mathbf{z}), \\ (z_n^2 - 1) Z(\overline{\mathfrak{S}}^C_{\lambda, s_n w}; \mathbf{z}) & = Z(\overline{\mathfrak{S}}^C_{\lambda, w}; \mathbf{z}) - Z(\overline{\mathfrak{S}}^C_{\lambda, w}; s_n \mathbf{z}). \end{align*} $$

We can also take $K = z_n^{-2} \cdot (z_n + 1) = z_n^{-1} + z_n^{-2}$ using the type $B \ K$ -matrix, which also is a solution to equation (3.4). In this case, we obtain

$$ \begin{align*} (z_n - 1) Z(\overline{\mathfrak{S}}^B_{\lambda, s_n w}; \mathbf{z}) & = Z(\overline{\mathfrak{S}}^B_{\lambda, w}; \mathbf{z}) - Z(\overline{\mathfrak{S}}^B_{\lambda, w}; s_n \mathbf{z}). \end{align*} $$

This is the functional equation for type B as desired.

Remark 3.9. We can construct another quasi-solvable lattice model $\overline {\mathfrak {S}}^R_{\lambda , w}$ by using the solution $K = -1$ to equation (3.4). In this case, our functional equation becomes

$$ \begin{align*} (z_n^2 - 1) Z(\overline{\mathfrak{S}}^R_{\lambda, s_n w}; s_n \mathbf{z}) & = Z(\overline{\mathfrak{S}}^R_{\lambda, w}; s_n \mathbf{z}) - Z(\overline{\mathfrak{S}}^R_{\lambda, w}; \mathbf{z}), \end{align*} $$

which is a flipped version of the type C functional equation.

3.3 The first main theorem

Using the functional equations we have shown, we can now prove our first main result.

Theorem 3.10. For Cartan type $X \in \{B, C\}$ , we have

$$\begin{align*}Z(\overline{\mathfrak{S}}^X_{\lambda, w}; \mathbf{z}) = \mathbf{z}^{\rho} A_w(\mathbf{z}, \lambda). \end{align*}$$

Proof. The case $w = 1$ is clear as $\mathbf {z}^{\rho } A_w(\mathbf {z}, \lambda ) = \mathbf {z}^{\lambda +\rho }$ by definition and there is a unique admissible state in $\overline {\mathfrak {S}}^X_{\lambda ,1}$ whose weight can be easily seen to be $\mathbf {z}^{\lambda +\rho }$ in both types B and C. The remainder of the proof proceeds by induction on the length of w and uses the functional equations proved in the previous sections.

Let $i < n$ with $j = i + 1$ , and let w be such that $\ell (s_iw) = \ell (w)+1$ . From our induction assumption, we have that $Z(\overline {\mathfrak {S}}_{\lambda , w}; \mathbf {z}) = \mathbf {z}^{\rho } A_w(\mathbf {z}, \lambda )$ and that

$$\begin{align*}Z(\overline{\mathfrak{S}}_{\lambda, w}; s_i \mathbf{z}) = z_i^{-1} z_j \cdot \mathbf{z}^{\rho} s_i A_w(\mathbf{z}, \lambda) = z_i^{-1} z_j \cdot \mathbf{z}^{\rho} A_w(s_i \mathbf{z}, \lambda). \end{align*}$$

Using the functional equation in Lemma 3.7, we obtain

$$\begin{align*}(z_i - z_j) Z(\overline{\mathfrak{S}}_{\lambda, s_i w}; \mathbf{z}) = z_j \cdot \mathbf{z}^{\rho} \bigl( A_w(\mathbf{z}, \lambda) - A_w(s_i \mathbf{z}, \lambda) \bigr), \end{align*}$$

which combined with equation (2.10a) produces $Z(\overline {\mathfrak {S}}_{\lambda , s_iw}; \mathbf {z}) = \mathbf {z}^{\rho } A_{s_iw}(\mathbf {z}, \lambda )$ as desired.

For $i = n$ , we note that $s_n \mathbf {z}^{\rho } = \mathbf {z}^{\rho }$ . It is then easy to see that by using the induction assumption together with Lemma 3.8 and equation (2.10b) in type B or equation (2.10c) in type C, we obtain that $Z(\overline {\mathfrak {S}}_{\lambda , s_n w}; \mathbf {z}) = \mathbf {z}^{\rho } A_{s_n w}(\mathbf {z}, \lambda )$ .

Example 3.11. Let $n = 2$ and $\lambda = (2,1)$ . Then we have the following states in $\overline {\mathfrak {S}}_w$ :

Next, we compute the partition functions for each model in type C for $\overline {Z}_w := Z(\overline {\mathfrak {S}}^C_{\lambda , w}; \mathbf {z})$ :

$$ \begin{gather*} \overline{Z}_1 = z_1^3 z_2, \qquad\qquad \overline{Z}_{s_1} = z_1^2 z_2^2, \qquad\qquad \overline{Z}_{s_2} = z_1^3 z_2^{-1}, \\ \overline{Z}_{s_1 s_2} = z_1^2 + z_2^2 + z_1 z_2, \qquad\qquad \overline{Z}_{s_2 s_1} = z_1^2 + z_1^2 z_2^{-2}, \\ \overline{Z}_{s_1 s_2 s_1} = 1 + z_1 z_2 + z_1 z_2^{-1} + z_1^{-1} z_2, \qquad\qquad \overline{Z}_{s_2 s_1 s_2} = 1 + z_2^{-2} + z_1 z_2^{-1}, \\ \overline{Z}_{w_0} = z_1^{-1} z_2^{-1}. \end{gather*} $$

We can see that these differ by $\mathbf {z}^{\rho } = z_1$ from the atoms $A_w := A_w(\mathbf {z}, \lambda )$ :

$$ \begin{gather*} A_1 = z_1^2 z_2, \qquad\qquad A_{s_1} = z_1 z_2^2, \qquad\qquad A_{s_2} = z_1^2 z_2^{-1}, \\ A_{s_1 s_2} = z_1 + z_1^{-1} z_2^2 + z_2, \qquad\qquad A_{s_2 s_1} = z_1 + z_1 z_2^{-2}, \\ A_{s_1 s_2 s_1} = z_1^{-1} + z_2 + z_2^{-1} + z_1^{-2} z_2, \qquad\qquad A_{s_2 s_1 s_2} = z_1^{-1} + z_1^{-1} z_2^{-2} + z_2^{-1}, \\ A_{w_0} = z_1^{-2} z_2^{-1}. \end{gather*} $$

3.4 Lattice models and quantum supergroups

Quantum (super)groups and solvable lattice models are naturally related by identifying the L-matrix or R-matrix of Boltzmann weights with R-matrices coming from representations of affine quantum (super)groups. In Proposition 3.12, we give a partial quantum supergroup interpretation of the solutions to the Yang–Baxter equation for $R^{\Gamma }_{\Gamma }$ and $R^{\Delta }_{\Delta }$ in terms of a $q \to 0$ limit of certain R-matrices related to representations of the quantum supergroup $U_q\bigl (\widehat {\mathfrak {gl}}(2n|1)\bigr )$ . However, after Proposition 3.12, we discuss why a complete quantum group interpretation is impossible in our setting. This is not surprising due to the lack of uniqueness of the R-matrix in Figure 8. Matching lattice model R-matrices to quantum group R-matrices has possible implications for future work relating lattice models with Iwahori Whittaker functions or Hall–Littlewood polynomials for symplectic or odd orthogonal groups. For example, in Cartan type A, a quantum group interpretation of the lattice model R-matrix can be used to relate quantum R-matrices with p-adic intertwining integrals [Reference Brubaker, Buciumas, Bump and Gustafsson15]. The $q=0$ results in this section (Proposition 3.12) are a first step towards developing a similar theory in Cartan type B and C.

Let $R_q := R_q(z_j/z_i)$ be the $U_q\bigl (\widehat {\mathfrak {gl}}(2n|1)\bigr ) \ R$ -matrix defined in [Reference Kojima45, Def. 2.1] acting on a tensor product of evaluation representations $V_{z_j} \otimes V_{z_i}$ , where $V_z$ is a $2n+1$ dimensional super vector space. We identify the basis of $V_z$ with the spins in our lattice model by the $2n$ colours corresponding to the even subspace and the $0$ for the odd subspace. We may write Boltzmann weights corresponding to the $R_q(z)$ matrix as in Figure 12. Denote by $R_q^{**}$ the $U_q\bigl (\widehat {\mathfrak {gl}}(2n|1)\bigr ) \ R$ -matrix acting on the dual of the evaluation representations $V_{q^2 z_j}^* \otimes V_{q^2 z_i}^*$ .

Figure 12 The $R_q$ -matrix in [Reference Kojima45] with $z = z_j/z_i$ , ${\color {red} c}> {\color {blue} c'}$ and ${\color {brown} d}$ being any colour.

Proof. To obtain $R^{\Gamma }_{\Gamma }$ in Figure 5 as the $q \to 0$ limit of $R_q$ in Figure 12, perform the following manipulations on $R_q$ , where $z = z_j / z_i$ :

1. 1. Multiply all fully coloured states by $-1$ (this corresponds to passing from a graded solution of the Yang–Baxter equation to an ungraded solution of the Yang–Baxter equation).

2. 2. Multiply weights $\mathtt {c}_1$ and $\mathtt {c}^{\prime }_1$ by $z^{-1}$ and weights $\mathtt {c}_2$ and $\mathtt {c}^{\prime }_2$ by z (this corresponds to a change of basis in $V_z$ and does not affect the quantum group structure).

3. 3. Multiply weight $\mathtt {b}_1$ by q and weight $\mathtt {b}_2$ by $q^{-1}$ ; multiply weight $\mathtt {b}^{\prime }_1$ by $-q$ and weight $\mathtt {b}^{\prime }_2$ by $-q^{-1}$ (this corresponds to a Drinfeld twist that affects the quantum group structure, namely its comultiplication, and universal R-matrix [Reference Drinfeld29, Reference Reshetikhin68]).

4. 4. Take the limit $q \to 0$ .

To compute the $q \to 0$ limit of $R_q^{**}$ , we use two standard facts from the theory of quantum groups. The first fact is that given a quantum group (more precisely, a quasitriangular Hopf algebra) representation V with $v \in V$ and its dual $V^*$ with $v^* \in V^*$ , then for any element h of the quantum group, we have that $h \cdot v^*(v) = v^* (S(h) \cdot v)$ , where S is the antipode. The second fact we will need is the property that the universal R-matrix $\mathcal {R}$ satisfies the relation

$$\begin{align*}(S \otimes S) \mathcal{R} = \mathcal{R}, \end{align*}$$

which is proved, for example, in [Reference Chari and Pressley24, Prop. 4.2.7]. These immediately imply that if $R^{\Gamma }_{\Gamma }$ is the $q \to 0$ limit of the $R_q$ , then $(R^{\Gamma }_{\Gamma })^t$ is the $q \to 0$ limit of the R-matrix $R_q^{**}$ .

Note that the R-matrix corresponding to $V_{z_j} \otimes V_{z_i}$ only depends on $z_j / z_i$ , therefore the R-matrices corresponding to $V^*_{z_j} \otimes V^*_{z_i}$ and $V^*_{q^2 z_j} \otimes V^*_{q^2 z_i}$ will be equal. What remains to show then is the relation $(R^{\Gamma }_{\Gamma })^t = R^{\Delta }_{\Delta }$ . This can be seen by comparing Figure 5 and Figure 6 and using the fact that taking transpose of an R-matrix modifies the states as follows:

To argue that we can interpret the horizontal edges in terms of $U_q\bigl (\widehat {\mathfrak {gl}}(2n|1)\bigr )$ -representations, we also need to examine $R^{\Delta }_{\Gamma }$ and $R^{\Gamma }_{\Delta }$ . The R-matrix $R^{\Gamma }_{\Delta }$ would correspond to the R-matrix $R_q^*$ associated to $V^*_{q^2z_1} \otimes V_{z_2}$ . One may compute this R-matrix by using

$$\begin{align*}(S \otimes \operatorname{id}) \mathcal{R} = \mathcal{R}^{-1}. \end{align*}$$

We also note that the R-matrix can be explicitly computed from [Reference Zhang80, Eq. (2.9)]. The $\mathtt {a}_1$ and $\mathtt {a}_2$ entries of this R-matrix will be (following Zhang [Reference Zhang80]) $bq-aq^{-1}$ and $bq^{-1}-aq$ . In trying to match these two entries with the corresponding entries in the $R^{\Gamma }_{\Delta }$ matrix, we can rescale both entries by a factor and set the parameters a and b. Taking the limit $q \to 0$ or $q\to \infty $ , we will not be able to match them with the factors $z_i -z_j$ and $z_j$ in a way that no factor will blow up or $\mathtt {b}_1 \neq 0$ .

Finally, we may consider the R-matrix $R^{\Delta }_{\Gamma }$ , which corresponds to the R-matrix associated to $V_{z_1} \otimes V^*_{q^2z_2}$ . We compute this quantum R-matrix explicitly for $n = 1,2,3$ by directly inverting the R-matrix given by [Reference Zhang80, Eq. (2.9)] (this uses the unitarity of the affine R-matrix). In particular, for $n = 2$ , the $\mathtt {a}_1$ and $\mathtt {a}_2$ entries of the R-matrix for $V_b \otimes V^*_a$ are $q^7 a - q^{-1} b$ and $q^5 a - q b$ , respectively. By a similar argument to the previous paragraph, we are also unable to obtain the desired $R^{\Delta }_{\Gamma }$ -matrix (see Figure 7).

The failure to simultaneously match the $\Gamma $ and $\Delta $ horizontal rows with representations of the quantum affine group $U_q\bigl ( \widehat {\mathfrak {gl}}(2n|1) \bigr )$ is in accordance with the nonuniqueness of the $R^{\Gamma }_{\Delta }$ matrix in Figure 8. If we had a match, we would expect to compute a unique $R^{\Gamma }_{\Delta }$ -matrix. A similar phenomena appears in the work of Zhong [Reference Zhong81].

One may also try to relate our K-matrix with affine K-matrices corresponding to a version of quantum symmetric pairs. Unfortunately, there is not much research on super versions of quantum affine pairs, so there is not much we can say on this subject.

5.1 Symplectic patterns and King tableaux

We consider the case for $G = \operatorname {Sp}_{2n}$ , which is the Lie group of Cartan type $C_n$ . We note that these patterns were first given by Želobenko [Reference Želobenko79]. A ${\color {darkred}{\textit{symplectic Proctor pattern}}}$ is a pattern of nonnegative integers of the form

$$\begin{align*}\begin{array}{cccccccccc} a_{1,1} && a_{1,2} && a_{1,3} && \cdots && a_{1,n} \\ & b_{1,1} && b_{1,2} && b_{1,3} && \cdots && b_{1,n} \\ && a_{2,2} && a_{2,3} && \cdots && a_{2,n} \\ &&& b_{2,2} && b_{2,3} && \cdots && b_{2,n} \\ &&&& \ddots && \ddots && \vdots & \vdots \\ &&&&&& a_{n-1,n-1} && a_{n-1,n} \\ &&&&&&& b_{n-1,n-1} && b_{n-1,n} \\ &&&&&&&& a_{n,n} \\ &&&&&&&&& b_{n,n} \end{array} \end{align*}$$

that satisfies the interlacing conditions

$$ \begin{align*} \min \{a_{i,j}, a_{i+1,j} \} & \geq b_{i,j} \geq \max \{a_{i,j+1}, a_{i+1,j+1} \}, \\ \min \{b_{i-1,j-1}, b_{i,j-1} \} & \geq a_{i,j} \geq \max \{b_{i-1,j}, b_{i,j} \}. \end{align*} $$

The weight of a symplectic Proctor pattern P is given by

$$\begin{align*}\operatorname{wt}(P) := \prod_{i=1}^n z_i^{A_i-2B_i+A_{i+1}}, \qquad \text{ where } A_i = \sum_{j=i}^n a_{i,j} \text{ and } B_i = \sum_{j=i}^n b_{i,j}. \end{align*}$$

We consider $A_{n+1} = 0$ . Let $\mathcal {P}^C_{\lambda }$ denote the set of symplectic Proctor patterns with top row $\lambda $ .

A ${\color {darkred}{\textit{King tableau}}}$ [Reference King43, Reference King44] is a filling of a Young diagram with entries in the ordered alphabet $1 < \overline {1} < 2 < \overline {2} < \cdots < n < \overline {n}$ such that the rows are weakly increasing and columns are strictly increasing and the smallest entry in row i is i. The weight of a King tableau T is

$$\begin{align*}\operatorname{wt}(T) = \prod_{i=1}^n x_i^{m_i - m_{\overline{\imath}}}, \end{align*}$$

where $m_k$ is the number of times k appears in T. Let $\mathcal {K}_{\lambda }$ denote the set of King tableaux of shape $\lambda $ . A ${\color {darkred}{\textit {reverse}}}$ King tableau is a King tableau with respect to the alphabet in the reverse order; or, alternatively, the entries in rows (respectively, columns) are weakly (respectively, strictly) decreasing and the largest entry in row i is $\overline {n + 1 - i}$ .

As discussed in [Reference Proctor66], there is a natural bijection $\Theta ^C \colon \mathcal {P}^C_{\lambda } \to \mathcal {K}_{\lambda }$ by extending the usual bijection between Gelfand–Tsetlin (GT) patterns and semistandard tableaux. Indeed, the partition $\lambda ^{(k)}$ of the kth row indicates the subtableau consisting of all of the letters greater than the kth letter in the alphabet. For instance, if $k = 3$ , then we restrict to the letters $\overline {n-1} < n < \overline {n}$ .

Proposition 5.1 [Reference Ivanov37, Ch. 1].

Let $\mathfrak {S}^C_{\lambda }$ denote the uncoloured model for $G = \operatorname {Sp}_{2n}$ . There exists a weight-preserving bijection

$$\begin{align*}\Psi^C \colon \mathfrak{S}^C_{\lambda} \to \mathcal{P}^C_{\lambda}. \end{align*}$$

Ivanov constructed the bijection $\Psi ^C$ in Proposition 5.1 explicitly by extending the usual bijection between the five-vertex model and GT patterns (see, for example, [Reference Buciumas, Scrimshaw and Weber20, Sec. 2.1]) and using the $1$ edges between the ${}^{\Delta }_{\Gamma }$ (respectively, ${}^{\Gamma }_{\Delta }$ ) rows to define the $\{a_{ij}\}_{i,j}$ (respectively, $\{b_{ij}\}_{i,j}$ ) values by the GT pattern bijection. More precisely, the ith row of vertical edges in the model is the $01$ -sequence of the partition in the ith row of the symplectic Proctor pattern read from right to left. We can make the analogous injection on the coloured model $\overline {\mathfrak {S}}_{\lambda , w}$ or $\mathfrak {S}_{\lambda , w}$ by considering the positions of the coloured vertical edges or equivalently by forgetting about the colours in our model as an intermediate step. Note that this is unaffected by the difference between the uncoloured version of our lattice model and that in [Reference Ivanov37] (see Remark 3.1).

Example 5.2. Consider the states given in Example 3.11 for $\overline {\mathfrak {S}}_{\lambda , w}$ . Then under the bijection $\Psi ^C$ , the states correspond to the following symplectic Proctor patterns:

$$ \begin{align*} 1: & \quad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 1 && 0 \\ && 1 \\ &&& 0 \end{array} \qquad s_1: \quad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 0 \\ && 2 \\ &&& 0 \end{array} \qquad s_2: \quad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 1 && 0 \\ && 1 \\ &&& 1 \end{array} \\ s_1 s_2: & \quad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 1 && 0 \\ && 0 \\ &&& 0 \end{array} \qquad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 1 \\ && 2 \\ &&& 0 \end{array} \qquad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 1 && 1 \\ && 1 \\ &&& 0 \end{array} \\ s_2 s_1: & \quad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 0 \\ && 2 \\ &&& 1 \end{array} \qquad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 0 \\ && 2 \\ &&& 2 \end{array} \\ s_1 s_2 s_1: & \quad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 0 \\ && 0 \\ &&& 0 \end{array} \qquad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 0 \\ && 1 \\ &&& 0 \end{array} \qquad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 0 \\ && 1 \\ &&& 1 \end{array} \qquad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 1 \\ && 1 \\ &&& 0 \end{array} \\ s_2 s_1 s_2: & \quad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 1 \\ && 2 \\ &&& 1 \end{array} \qquad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 1 \\ && 2 \\ &&& 2 \end{array} \qquad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 1 && 1 \\ && 1 \\ &&& 1 \end{array} \\ w_0: & \quad \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 2 && 1 \\ & 2 && 1 \\ && 1 \\ &&& 1 \end{array} \end{align*} $$

5.2 Odd orthogonal patterns and Sundaram tableaux

Here we instead assume $G = \operatorname {SO}_{2n+1}$ , which is the Lie group of Cartan type $B_n$ . An ${\color {darkred}{\textit {odd}}} {\color {darkred}{\textit {orthogonal Proctor pattern}}}$ is a symplectic Proctor pattern such that the values $b_{i,n}$ , for all $1 \leq i \leq n$ , at the right ends are also allowed to be half integers. We remark that these patterns were first announced by Gelfand and Tsetlin without proof in [Reference Gel’fand and Tsetlin34]. Let $\mathcal {P}^B_{\lambda }$ denote the set of odd orthogonal Proctor patterns with top row $\lambda $ .

A Sundaram tableau [Reference Sundaram73] is a King tableau with an additional letter $\infty> \overline {n}$ that is allowed to repeat down columns but can only appear once in a row. The weight of a Sundaram tableau is the same as for a King tableau; in particular, we ignore $\infty $ in the weight computation. We denote the set of Sundaram tableau of shape $\lambda $ by $\mathcal {S}_{\lambda }$ . A reverse Sundaram tableau is defined analogously to a reverse King tableau. Likewise, we have a natural bijection $\Theta ^B \colon \mathcal {P}^B_{\lambda } \to \mathcal {S}_{\lambda }$ , as noted in [Reference Proctor66], by the same description as $\Theta ^C$ , except that if the rightmost entry in the odd orthogonal Proctor pattern is a half integer, we replace the leftmost entry in the corresponding row with an $\infty $ .

Recall that the models for both types B and C Demazure atoms (as well as for Demazure characters) have the same states, but the $\mathtt {k}_1$ entry of the K-matrix has a binomial weight $z^{-2} + z^{-1}$ in type B as opposed to the monomial weight $z^{-2}$ for type C. Thus, following [Reference Buciumas, Scrimshaw and Weber20], we can introduce a marking to the states for this K-matrix entry where if the bend is marked, then it has a Boltzmann weight of $z^{-1}$ ; otherwise the Boltzmann weight is $z^{-2}$ . This yields a bijection between the marked states and the monomials of the partition function as opposed to a product of binomials.

Proposition 5.3. Let $\widehat {\mathfrak {S}}_{\lambda }$ denote the set of marked states for the uncoloured type B model. There exists a weight-preserving bijection

$$\begin{align*}\Psi^B \colon \widehat{\mathfrak{S}}_{\lambda} \to \mathcal{P}^B_{\lambda}. \end{align*}$$

Similar to the case when $X = C$ , we can extend $\Psi ^B$ to an injection with the domain the coloured model $\overline {\mathfrak {S}}_{\lambda , w}^B$ or $\mathfrak {S}^B_{\lambda , w}$ .

Example 5.4. Let us take $w = s_2$ . Then we have the one state in the model $\overline {\mathfrak {S}}^B_{\lambda , w}$ with one $\mathtt {k}_1$ U-turn that we can mark, which corresponds to the following pair of odd orthogonal Proctor patterns and reverse Sundaram tableaux

where the box in the unique state of $\overline {\mathfrak {S}}^B_{s_2,\lambda }$ denotes the possible marking.

5.3 Key algorithm

Using the bijections $\Psi ^X$ , we now give a simple algorithm for computing the (right) key w of a reverse King (respectively, Sundaram) tableau T for $X = C$ (respectively, $X = B$ ) using the bijections from [Reference Proctor66].

Let $\mathcal {T}$ denote a set of tableaux such that $\chi _{\lambda }(\mathbf {z}) = \sum _{T \in \mathcal {T}} \mathbf {x}^{\operatorname {wt}(T)}$ . The ${\color {darkred}{\textit {(right) key}}}$ of $\mathcal {T}$ is a map $\operatorname {\mathbf {key}} \colon \mathcal {T} \to W$ such that

$$\begin{align*}\mathcal{T} = \bigsqcup_{w \in W} \{T \in \mathcal{T} \mid \operatorname{\mathbf{key}}(T) = w\} \qquad \text{ and } \qquad A_w(\mathbf{z}, \lambda) = \sum_{\substack{T \in \mathcal{T} \\ \operatorname{\mathbf{key}}(T) = w}} \mathbf{z}^{\operatorname{wt}(T)}. \end{align*}$$

We consider the case when $\mathcal {T}$ is the set of reverse King (respectively, Sundaram) tableaux of shape $\lambda $ for $X = C$ (respectively, $X = B$ ). Consider some $T \in \mathcal {T}$ , which we convert first to the corresponding type of Proctor pattern and then to a state in the uncoloured lattice model. That is, the state we have is $S = \bigl ( (\Psi ^X)^{-1} \circ (\Theta ^X)^{-1} \bigr )(T)$ . From Theorem 4.5, Theorem 3.10 and the fact that $D_{w_0}(\mathbf {z}, \lambda ) = \chi _{\lambda }(\mathbf {z})$ , we know there is a unique way to colour the corresponding state S. This colouring gives us a signed permutation w by reading the left boundary of the coloured state, which then defines $\operatorname {\mathbf {key}}(T) = w$ .

For the remainder of this section, we assume $X = C$ unless stated otherwise. We give a conjecture that would justify that the key map given above is natural. We first need some notation from crystal theory. For more details on crystals, we refer the reader to [Reference Bump and Schilling22]. A highest-weight crystal $B(\lambda )$ is a certain edge-colored weighted connected digraph that satisfies certain additional properties and encodes the action of the corresponding quantum group of G. In particular, we have operators $e_i$ defined by $e_i(b) = b'$ for every edge $b' \xrightarrow [\hspace {20pt}]{i} b$ . Let $u_{\lambda }$ denote the highest-weight element of $B(\lambda )$ , which is the unique source in the digraph.

Fix some $w \in W$ , and choose some reduced expression $w = s_{i_1} s_{i_2} \dotsm s_{i_{\ell }}$ . A ${\color {darkred}{\textit {Demazure crystal}}}$ is the crystal induced from $B(\lambda )$ by the set of vertices

$$\begin{align*}B_w(\lambda) := \{ b \in B(\lambda) \mid e_{i_{\ell}}^{a_{\ell}} \cdots e_{i_2}^{a_2} e_{i_1}^{a_1} b = u_{\lambda} \text{ for some } a_1, \dotsc, a_{\ell} \in \mathbb{Z}_{\geq 0} \}. \end{align*}$$

This does not depend on the choice of reduced expression for w [Reference Kashiwara40]. The ${\color {darkred}{\textit {atom crystal}}}$ is the crystal induced from $B(\lambda )$ by the set of vertices

$$\begin{align*}\overline{B}_w(\lambda) := B_w(\lambda) \setminus \bigcup_{v < w} B_v(\lambda). \end{align*}$$

A common model for $B(\lambda )$ for type $C_n$ is the set of Kashiwara–Nakashima (KN) tableaux, which has a natural crystal structure [Reference Kashiwara and Nakashima41].

The Sheats bijection [Reference Sheats72] is a weight-preserving bijection between King tableaux and a variant of KN tableaux called DeConcini tableaux [Reference De Concini26]. There is a natural map from KN tableaux to DeConcini tableaux, so we will abuse terminology slightly and call the composite of these maps the Sheats bijection from KN tableaux to King tableaux. However, we have the weight twisting by $i \leftrightarrow \overline {n+1-i}$ coming from the bijection from KN tableaux to DeConcini tableaux. To go between reverse King tableaux and normal King tableaux, we simply replace $i \leftrightarrow \overline {n+1-i}$ entry-wise, where $\overline {\overline {\imath }} = i$ . To get the alphabets to match, we require interchanging $i \leftrightarrow \overline {\imath }$ by using the tableau switching algorithm from [Reference Benkart, Sottile and Stroomer9]. This leads to the following conjecture.

We note that first two maps (1) and (2) commute.

Example 5.7. Consider the states given in Example 3.11 and the symplectic Proctor patterns under $\Psi ^C$ in Example 5.2. The corresponding reverse King tableaux with the alphabet $1 < \overline {1} < 2 < \overline {2}$ are

Next we change the alphabet by $i \leftrightarrow \overline {\imath }$ using the tableau-switching algorithm, so our new alphabet is $\overline {1} < 1 < \overline {2} < 2$ , to yield

We then apply the map that sends $i \leftrightarrow \overline {n+1-i}$ entry-wise to obtain

Finally, we use the Sheats bijection to obtain the KN tableaux (with the alphabet now $1 < 2 < \overline {2} < \overline {1}$ )

which are precisely the crystal atoms (see [Reference Santos71, Ex. 16]).

We remark that applying $w_0$ to the weight means we have $w_0 \operatorname {wt} = -\operatorname {wt}$ , and in terms of the bijection with Proctor patterns, we instead use the initial alphabet $\overline {1} < 1 < \overline {2} < 2$ . In this case, we are taking dual atoms from the lowest weight element and replacing $e_i \mapsto f_i$ in the definition of a Demazure crystal. Equivalently, we are applying the Lusztig involution (or the contragradient dual) to the crystal atom. However, this does not remove the fact we are working with reverse King tableaux unlike in [Reference Buciumas, Scrimshaw and Weber20].

Example 5.8. We note that applying tableau switching $i \leftrightarrow \overline {\imath }$ is not the same as taking the Proctor patterns for the alphabet changed under $w_0$ . Indeed, the following tableaux are fixed under the tableau switching but are interchanged using the alphabet $\overline {1} < 1 < \overline {2} < 2$ :

If we instead took the order of the spectral parameters from [Reference Gray35, Reference Ivanov37], the weight would instead be twisted by the longest element $w_A$ of natural $S_n \subseteq W$ . Yet this would give us King tableaux as our image under the bijection $\Phi ^C$ . This would mean we would have a key algorithm on King tableaux for a twisted version of atoms working with an extremal weight crystal $B(w_A \lambda )$ in the terminology of [Reference Kashiwara40]. We have similar results for $G = \operatorname {SO}_{2n+1}$ and Sundaram tableaux, as well as conjecture a crystal structure that is an extension of a solution of Conjecture 5.6.