A q-boson representation of Zamolodchikov-Faddeev algebra for stochastic R matrix of $\varvec{U_q(A^{(1)}_n)}$

Abstract

We construct a q-boson representation of the Zamolodchikov-Faddeev algebra whose structure function is given by the stochastic R matrix of $U_q(A^{(1)}_n)$ introduced recently. The representation involves quantum dilogarithm type infinite products in the $n(n-1)/2$-fold tensor product of q-bosons. It leads to a matrix product formula of the stationary probabilities in the $U_q(A_n^{(1)})$-zero range process on a one-dimensional periodic lattice.

Appendix A: Proof of Theorem 1

We will use the same symbol $\varphi (\beta ,\gamma )$ (3) to mean (3)$|_{n \rightarrow n+1}$. Moreover, $\sum _{1\le i < j \le n+1}\beta _i \gamma _j$ with $\beta \in {\mathbb Z}_{\ge 0}^n, \gamma \in {\mathbb Z}_{\ge 0}^{n+1}$ will also be denoted by $\varphi (\beta ,\gamma )$.

We are going to prove Theorem 1 by induction on n. The relation (15)$|_{n=1}$ is valid since it is equivalent to (9)$|_{n=1}$. (The case $n=2$ has been shown in [16] by a method different from here.) Thus, our task in the sequel is to show (15)$|_{n\rightarrow n+1}$ assuming (15)$|_{n=n}$.

Lemma 7

Under the assumption (15) $|_{n=n}$, the relation (15) $|_{n\rightarrow n+1}$ follows from the equality

$$\begin{aligned}&\sum \limits _{m \le s}q^{\varphi (m,\alpha )}\Phi ^{(n)}_q(m|s; \lambda , \mu ) \bigotimes \limits _{i=1}^n\mathbf{b}^{s_i-m_i}\mathbf{c}^{\alpha _i}\mathbf{b}^{m_i}\nonumber \\&\quad = \sum \limits _{\gamma \le \alpha }q^{\varphi (\gamma ,\alpha )+ \varphi (s,\gamma )-\varphi (\alpha ,\gamma )} \Phi ^{(n+1)}_q(\gamma |\alpha , \lambda , \mu )\bigotimes \limits _{i=1}^n \mathbf{c}^{\gamma _i}\mathbf{b}^{s_i}\mathbf{c}^{\alpha _i-\gamma _i}\; (\forall s \in {\mathbb Z}_{\ge 0}^n, \forall \alpha \in {\mathbb Z}_{\ge 0}^{n+1}). \end{aligned}$$

(39)

Proof

Substitute (19)$|_{n \rightarrow n+1}$ into (15)$|_{n\rightarrow n+1}$. Applying (13) on the LHS, we get

$$\begin{aligned} \text {LHS}&= \sum _{m,l \in {\mathbb Z}_{\ge 0}^n}\sum _{t \le m} \Phi ^{(n)}_q(l|m+l-t;\lambda ,\mu ) X_t(\lambda )X_{m+l-t}(\mu )\\&\quad \otimes \bigotimes _{i=1}^n(m_i,\alpha ^+_i,\alpha _i)(l_i, \beta ^+_i, \beta _i),\\ \text {RHS}&= \sum _{\gamma \le \alpha } q^{\varphi (\alpha -\gamma ,\beta -\gamma )} \Phi ^{(n+1)}_q(\gamma |\alpha , \lambda , \mu ) \sum _{t,s \in {\mathbb Z}_{\ge 0}^n} X_t(\lambda )X_s(\mu ) \otimes \bigotimes _{i=1}^n (t_i, \gamma ^+_i, \gamma _i)\\&\quad \times (s_i, \alpha ^+_i+\beta ^+_i-\gamma ^+_i, \alpha _i+\beta _i-\gamma _i), \end{aligned}$$

where the temporal notation $(i,j,k) := \mathbf{b}^i \mathbf{k}^j \mathbf{c}^k$ is used. The symbol $\alpha ^+_i$ is defined by (17)$|_{n \rightarrow n+1}$ and $\beta ^+_i, \gamma ^+_i$ are similar. Thus, to prove $\text {LHS}=\text {RHS}$, it is sufficient, though not a priori necessary, to show that the coefficients of $X_t(\lambda )X_s(\mu )$ are equal for each choice of $t,s \in {\mathbb Z}_{\ge 0}^n$ and $\alpha , \beta \in {\mathbb Z}_{\ge 0}^{n+1}$. Explicitly, it reads

$$\begin{aligned} \begin{array}{ll} &{}\sum \limits _{t \le m \le t+s}\Phi ^{(n)}_q(t+s-m|s; \lambda , \mu ) \bigotimes \limits _{i=1}^n(m_i,\alpha ^+_i,\alpha _i)(t_i+s_i-m_i, \beta ^+_i, \beta _i)\\ &{} \qquad =\sum \limits _{\gamma \le \alpha }q^{\varphi (\alpha -\gamma ,\beta -\gamma )} \Phi ^{(n+1)}_q(\gamma |\alpha , \lambda , \mu ) \bigotimes \limits _{i=1}^n (t_i, \gamma ^+_i, \gamma _i) \\ &{} \quad \qquad \times (s_i, \alpha ^+_i+\beta ^+_i-\gamma ^+_i, \alpha _i+\beta _i-\gamma _i). \end{array} \end{aligned}$$

One can pull out the common factor $\bigotimes _{i=1}^n\mathbf{k}^{\alpha ^+_i+\beta ^+_i}$ from this equality to the left. The result reads

$$\begin{aligned} \begin{array}{ll} &{}\sum \limits _{t \le m \le t+s}q^{\varphi (t+s-m,\alpha )} \Phi ^{(n)}_q(t+s-m|s; \lambda , \mu ) \bigotimes \limits _{i=1}^n(m_i,0,\alpha _i)(t_i+s_i-m_i,0, \beta _i)\\ &{}\qquad = \sum \limits _{\gamma \le \alpha }q^{\varphi (\gamma ,\alpha )+ \varphi (s,\gamma )-\varphi (\alpha ,\gamma )} \Phi ^{(n+1)}_q(\gamma |\alpha , \lambda , \mu ) \bigotimes \limits _{i=1}^n (t_i, 0, \gamma _i)(s_i, 0, \alpha _i+\beta _i-\gamma _i). \end{array} \end{aligned}$$

Note that this further contains a common rightmost factor $\bigotimes _{i=1}^n \mathbf{c}^{\beta _i}$ and a common leftmost factor $\bigotimes _{i=1}^n\mathbf{b}^{t_i}$. Removing them leads to (39).$\square$

Lemma 8

The following identities hold:

$$\begin{aligned} \mathbf{c}^m\mathbf{b}^s&= \sum _{j=0}^s q^{j(m-s+j)}(q^m;q^{-1})_{s-j}\left( {\begin{array}{c}s\\ j\end{array}}\right) _{q} \mathbf{b}^j\mathbf{c}^{m-s+j} \qquad (m,s \in {\mathbb Z}_{\ge 0}), \end{aligned}$$

(40)

$$\begin{aligned} z^s&=\sum _{r=0}^s (-1)^r q^{r(r-1)/2}\left( {\begin{array}{c}s\\ r\end{array}}\right) _{q}(z;q^{-1})_r \qquad (s \in {\mathbb Z}_{\ge 0}). \end{aligned}$$

(41)

Proof

These relations can easily be checked by means of the q-binomial theorem.$\square$

Lemma 9

The equality (39) is equivalent to

$$\begin{aligned}&\sum _{m \le s} q^{\varphi (m,\alpha )+ \sum _{i=1}^n\alpha _im_i} \Phi ^{(n)}_q(m|s;\lambda ,\mu ) \nonumber \\&\qquad = \sum _{\gamma \le \alpha } q^{\varphi (\gamma ,\alpha )+\varphi (s,\gamma )- \varphi (\alpha ,\gamma )+\sum _{i=1}^ns_i\gamma _i} \Phi ^{(n+1)}_q(\gamma |\alpha ;\lambda ,\mu ) \end{aligned}$$

(42)

for any $s \in {\mathbb Z}_{\ge 0}^n$ and $\alpha \in {\mathbb Z}_{\ge 0}^{n+1}$.

Proof

Comparing the coefficient of the basis vector $\bigotimes _{i=1}^n\mathbf{b}^{s_i-p_i}\mathbf{c}^{\alpha _i-p_i}$ of $\mathcal {B}^{\otimes n}$ on the both sides of (39) by means of (40), it is translated into the equality of the coefficients

$$\begin{aligned}&\sum _{m \le s}q^{\varphi (m,\alpha )} \Phi ^{(n)}_q(m|s;\lambda ,\mu ) \prod _{i=1}^n q^{(m_i-p_i)(\alpha _i-p_i)} \left( {\begin{array}{c}\alpha _i\\ p_i\end{array}}\right) _{q}\left( {\begin{array}{c}m_i\\ p_i\end{array}}\right) _{q}\nonumber \\&\qquad = \sum _{\gamma \le \alpha }q^{\varphi (\gamma ,\alpha ) +\varphi (s^{\vee }-\alpha ,\gamma )} \Phi ^{(n+1)}_q(\gamma |\alpha ;\lambda ,\mu ) \prod _{i=1}^n q^{(\gamma _i-p_i)(s_i-p_i)} \left( {\begin{array}{c}s_i\\ p_i\end{array}}\right) _{q}\left( {\begin{array}{c}\gamma _i\\ p_i\end{array}}\right) _{q} \end{aligned}$$

(43)

for any arrays of nonnegative integers $\alpha =(\alpha _1, \ldots , \alpha _n,\alpha _{n+1}), s=(s_1,\ldots , s_n), p=(p_1,\ldots , p_n)$ such that $p_i \le \min (s_i,\alpha _i)$ for all $1 \le i \le n$. On the RHS of (43), we have introduced the notation

$$\begin{aligned} s^{\vee } = (s_1,\ldots , s_n,0) \end{aligned}$$

(44)

for later convenience. Of course $\varphi (s^{\vee },\gamma ) = \varphi (s,\gamma )$ by the definition. By substituting (3) into (43) and removing a common overall factor from the both sides, it becomes

$$\begin{aligned} \begin{array}{ll} &{}\sum \limits _{p \le m \le s}q^{\varphi (m,\alpha -m^{\vee })+\varphi (s,m)} \nu ^{|m|}\frac{(\lambda )_{|m|}(\nu )_{|s|-|m|}}{(\mu )_{|s|}} \prod \limits _{i=1}^n q^{(m_i-p_i)(\alpha _i-p_i)}\left( {\begin{array}{c}s_i-p_i\\ m_i-p_i\end{array}}\right) _{q}\\ &{} \qquad = \sum \limits _{p^{\vee } \le \gamma \le \alpha }q^{ \varphi (\gamma ,\alpha -\gamma )+\varphi (s, \gamma )} \nu ^{|\gamma |}\frac{(\lambda )_{|\gamma |}(\nu )_{|\alpha |-|\gamma |}}{(\mu )_{|\alpha |}}\left( {\begin{array}{c}\alpha _{n+1}\\ \gamma _{n+1}\end{array}}\right) _{q} \prod \limits _{i=1}^n q^{(s_i-p_i)(\gamma _i-p_i)}\left( {\begin{array}{c}\alpha _i-p_i\\ \gamma _i-p_i\end{array}}\right) _{q}, \end{array} \end{aligned}$$

where $\nu = \mu /\lambda$ and $m^{\vee }, p^{\vee }$ are defined similar to (44). By the replacement

$$\begin{aligned}&s \rightarrow s+p,\;\; m \rightarrow m+p,\;\; \alpha \rightarrow \alpha +p^{\vee },\;\; \gamma \rightarrow \gamma +p^{\vee },\\&\quad \lambda \rightarrow q^{-|p|}\lambda ,\;\; \mu \rightarrow q^{-|p|}\mu , \end{aligned}$$

the above relation is cast into

$$\begin{aligned}&\sum \limits _{m \le s}q^{\varphi (m,\alpha )+\varphi (s-m,m)+\sum _{i=1}^n\alpha _im_i} \nu ^{|m|}\frac{(\lambda )_{|m|}(\nu )_{|s|-|m|}}{(\mu )_{|s|}} \prod \limits _{i=1}^n\left( {\begin{array}{c}s_i\\ m_i\end{array}}\right) _{q}\nonumber \\&\qquad =\sum \limits _{\gamma \le \alpha } q^{\varphi (\gamma ,\alpha ) +\varphi (s^{\vee }-\gamma ,\gamma )+\sum _{i=1}^ns_i\gamma _i} \nu ^{|\gamma |}\frac{(\lambda )_{|\gamma |} (\nu )_{|\alpha |-|\gamma |}}{(\mu )_{|\alpha |}} \prod \limits _{i=1}^{n+1}\left( {\begin{array}{c}\alpha _i\\ \gamma _i\end{array}}\right) _{q}, \end{aligned}$$

(45)

which turns out to be free from $p=(p_1,\ldots , p_n)$. This coincides with (42).$\square$

So far we have shown that Theorem 1 is a corollary of (42). Let us proceed to a proof of the latter.

Lemma 10

The equality (42) holds for $n \in {\mathbb Z}_{\ge 0}$.

Proof

Again we invoke the induction on n. At $n=0$, (42) reads $1= \sum _{\gamma _1 \le \alpha _1} \Phi ^{(1)}_q(\gamma _1|\alpha _1;\lambda , \mu )$, which is indeed valid thanks to (9). Assume (42)$|_{n\rightarrow n-1}$. Applying (11) to the LHS of (42)$|_{n=n}$, we get

$$\begin{aligned} \text {LHS} = \sum _{m_1 \le s_1} q^{m_1|\alpha |}\Phi ^{(1)}_q(m_1|s_1;\lambda , \mu ) \sum _{\overline{m} \le \overline{s}} q^{\varphi (\overline{m},\overline{\alpha }) + \sum _{i=2}^n m_i\alpha _i} \Phi ^{(n-1)}_q(\overline{m}|\overline{s}; q^{m_1}\lambda , q^{s_1}\mu ), \end{aligned}$$

where $\overline{m}, \overline{s} \in {\mathbb Z}_{\ge 0}^{n-1}$ and $\overline{\alpha } \in {\mathbb Z}_{\ge 0}^n$ are defined by (10). Rewriting the sum over $\overline{m}$ by the induction assumption (42)$|_{n\rightarrow n-1}$ yields ($\nu =\mu /\lambda$ as before)

$$\begin{aligned} \text {LHS}&= \sum _{m_1 \le s_1} q^{m_1|\alpha |}\Phi ^{(1)}_q(m_1|s_1;\lambda , \mu ) \nonumber \\&\quad \times \sum _{\overline{\gamma } \le \overline{\alpha }} q^{\varphi (\overline{\gamma },\overline{\alpha }) +\varphi (\overline{s},\overline{\gamma }) -\varphi (\overline{\alpha },\overline{\gamma })+\sum _{i=2}^ns_i\gamma _i} \Phi ^{(n)}_q(\overline{\gamma }|\overline{\alpha }; q^{m_1}\lambda , q^{s_1}\mu ) \nonumber \\&=\sum _{m_1\le s_1}q^{m_1(|\alpha |-|\overline{\gamma }|)} \nu ^{m_1}\left( {\begin{array}{c}s_1\\ m_1\end{array}}\right) _{q}\;\, \nonumber \\&\quad \times \sum _{\overline{\gamma } \le \overline{\alpha }} q^{\xi (s,\overline{\alpha }, \overline{\gamma })} \nu ^{|\overline{\gamma }|} \frac{(\lambda )_{m_1+|\overline{\gamma }|} (\nu )_{m_1-s_1+|\overline{\alpha }|-|\overline{\gamma }|}}{(\mu )_{s_1+|\overline{\alpha }|}} \prod _{i=2}^{n+1}\left( {\begin{array}{c}\alpha _i\\ \gamma _i\end{array}}\right) _{q}, \end{aligned}$$

(46)

where $\xi (s,\overline{\alpha }, \overline{\gamma }) =\varphi (\overline{\gamma }, \overline{\alpha })+\varphi (\overline{s},\overline{\gamma }) +\sum _{i=2}^ns_i\gamma _i -\varphi (\overline{\gamma },\overline{\gamma })+ s_1|\overline{\gamma }|$. On the other hand, the RHS of (42) has been written out in the RHS of (45), which is expressed using the above $\xi (s,\overline{\alpha }, \overline{\gamma })$ as

$$\begin{aligned} \text {RHS} = \sum _{\gamma _1 \le \alpha _1, \overline{\gamma } \le \overline{\alpha }} q^{\gamma _1(|\overline{\alpha }|-|\overline{\gamma }|+s_1)+ \xi (s,\overline{\alpha }, \overline{\gamma })} \nu ^{|\gamma |} \frac{(\lambda )_{|\gamma |} (\nu )_{|\alpha |-|\gamma |}}{(\mu )_{|\alpha |}} \prod _{i=1}^{n+1}\left( {\begin{array}{c}\alpha _i\\ \gamma _i\end{array}}\right) _{q}. \end{aligned}$$

(47)

Denote the summand in (46) by $\text {LHS}(m_1,\gamma _2,\ldots , \gamma _{n+1})$ and the one in (47) by $\text {RHS}(\gamma _1,\gamma _2,\ldots , \gamma _{n+1})$. We claim $\sum _{m_1\le s_1} \text {LHS}(m_1,\gamma _2,\ldots , \gamma _{n+1}) = \sum _{\gamma _1 \le \alpha _1} \text {RHS}(\gamma _1,\gamma _2,\ldots , \gamma _{n+1})$ holds for each fixed $\overline{\gamma } = (\gamma _2,\ldots , \gamma _{n+1})$. In fact, the two sides possess a common overall factor $q^{\xi (s,\overline{\alpha }, \overline{\gamma })} \nu ^{|\overline{\gamma }|} \frac{(\lambda )_{|\overline{\gamma }|} (\nu )_{|\overline{\alpha }|-|\overline{\gamma }|}}{(\mu )_{|\overline{\alpha }|}} \prod _{i=2}^{n+1}\left( {\begin{array}{c}\alpha _i\\ \gamma _i\end{array}}\right) _{q}$. By removing it, the claim becomes

$$\begin{aligned}&\sum _{m_1\le s_1} q^{m_1(|\alpha |-|\overline{\gamma }|)}\nu ^{m_1} \frac{(q^{|\overline{\gamma }|}\lambda )_{m_1} (q^{|\overline{\alpha }|-|\overline{\gamma }|}\nu )_{m_1-s_1}}{(q^{|\overline{\alpha }|}\mu )_{s_1}} \left( {\begin{array}{c}s_1\\ m_1\end{array}}\right) _{q}\\&\qquad =\sum _{\gamma _1 \le \alpha _1} q^{\gamma _1(|\overline{\alpha }|-|\overline{\gamma }|+s_1)} \nu ^{\gamma _1} \frac{(q^{|\overline{\gamma }|}\lambda )_{\gamma _1} (q^{|\overline{\alpha }|-|\overline{\gamma }|}\nu )_{\alpha _1-\gamma _1}}{(q^{|\overline{\alpha }|}\mu )_{\alpha _1}} \left( {\begin{array}{c}\alpha _1\\ \gamma _1\end{array}}\right) _{q}. \end{aligned}$$

This is simply stated as $f(\alpha _1, s_1; q^{|\overline{\gamma }|}\lambda , q^{|\overline{\alpha }|}\mu ) = f(s_1, \alpha _1; q^{|\overline{\gamma }|}\lambda , q^{|\overline{\alpha }|}\mu )$ in terms of the function defined for $s,t \in {\mathbb Z}_{\ge 0}$ and $\nu =\mu /\lambda$ by

$$\begin{aligned} f(s,t;\lambda ,\mu ) = \sum _{i=0}^t q^{si} \nu ^i \frac{(\lambda )_i(\nu )_{t-i}}{(\mu )_t}\left( {\begin{array}{c}t\\ i\end{array}}\right) _{q}. \end{aligned}$$

(48)

This is verified in Lemma 11.$\square$

Lemma 11

The function (48) enjoys the symmetry $f(s,t;\lambda ,\mu ) = f(t,s;\lambda ,\mu )$ for $s,t \in {\mathbb Z}_{\ge 0}$.

Proof

By applying (41) to the factor $q^{si}$ in (48), $f(s,t;\lambda , \mu )$ is rewritten as follows:

$$\begin{aligned} f(s,t;\lambda , \mu )&= \sum _{i=0}^t \nu ^i \frac{(\lambda )_i(\nu )_{t-i}}{(\mu )_t}\left( {\begin{array}{c}t\\ i\end{array}}\right) _{q} \sum _{r=0}^s(-1)^rq^{r(r-1)/2}\left( {\begin{array}{c}s\\ r\end{array}}\right) _{q}(q^i;q^{-1})_r\\&=\sum _{r=0}^{\min (s,t)}\sum _{i=r}^t \nu ^i \frac{(\lambda )_r(q^r\lambda )_{i-r}(\nu )_{t-i}}{(\mu )_r(q^r\mu )_{t-r}} (q^t;q^{-1})_r\left( {\begin{array}{c}t-r\\ i-r\end{array}}\right) _{q} (-1)^rq^{r(r-1)/2}\left( {\begin{array}{c}s\\ r\end{array}}\right) _{q}. \end{aligned}$$

Replacing i by $i+r$, we have

$$\begin{aligned} f(s,t;\lambda , \mu )&= \sum _{r=0}^{\min (s,t)} \nu ^r(-1)^rq^{r(r-1)/2}(q)_r\left( {\begin{array}{c}s\\ r\end{array}}\right) _{q}\left( {\begin{array}{c}t\\ r\end{array}}\right) _{q} \frac{(\lambda )_r}{(\mu )_r}h(r,t;\lambda , \mu ), \nonumber \\ h(r,t;\lambda , \mu )&= \sum _{i=0}^{t-r} \nu ^i \frac{(q^r\lambda )_{i}(\nu )_{t-r-i}}{(q^r\mu )_{t-r}}\left( {\begin{array}{c}t-r\\ i\end{array}}\right) _{q}. \end{aligned}$$

(49)

From $\sum _{i=0}^{t-r} \Phi ^{(1)}_q(i|t-r;q^r\lambda , q^r\mu )=1$ (9), we find $h(r,t;\lambda , \mu )=1$. Then the expression (49) tells that $f(s,t;\lambda , \mu )=f(t,s;\lambda , \mu )$.$\square$

Proof of Theorem 1

As a summary of the arguments so far, the induction step from (15)$|_{n=n}$ to (15)$|_{n=n+1}$ has been established by the following scheme:

Since (15)$|_{n=1}$ is valid as explained in the beginning of the appendix, the ZF relation (15) holds for any n. This completes the proof of Theorem 1. $\square$