Integrable boundary conditions for multi-species ASEP

We start by recalling the dynamical rules of the multi-species ASEP. We will call the model multi-species ASEP, with the convention that we consider a model with $(N-1)$ species of particles on a one-dimensional lattice with L sites. Each site on the lattice is occupied by a single particle, or is empty, and we identify this vacancy (or hole) as an additional species, that we call 1. The species of particles shall be labeled $2,3,\ldots ,N$. A configuration on the lattice is thus a L-tuplet $({\tau }_{1},{\tau }_{2},\ldots ,{\tau }_{L})$ that belongs to $\{1,\ldots ,N\}{}^{L}$. To each of the L sites we will associate a ${{\mathbb{C}}}^{N}$ vector space, so that the set of all configurations is embedded into the tensor space $\underset{L}{\underbrace{{{\mathbb{C}}}^{N}\otimes \cdots \,\otimes \,{{\mathbb{C}}}^{N}}}$. The natural basis of this space is given by $| {\tau }_{1}\rangle \otimes \cdots \,\otimes \,| {\tau }_{L}\rangle$ with ${\tau }_{i}=1,2,\ldots ,N$ and $| \tau \rangle ={(\underset{\tau -1}{\underbrace{0,\ldots ,0}},1,\underset{N-\tau }{\underbrace{0,\ldots ,0}})}^{t}$.

In the bulk. The dynamics is defined as follows: a bond $(i,i+1)$, with $1\leqslant i\leqslant L-1$, between two neighboring lattice sites, is updated between time t and t + dt by swapping the particles at i and $i+1$ with rate 1 or q depending on the local configuration ${\tau }_{i}\,{\tau }_{i+1}$ involved

$$\begin{eqnarray}&&{\tau }_{i}\,{\tau }_{i+1}\,\overset{1}{\to }\,{\tau }_{i+1}\,{\tau }_{i}\quad {\rm{if}}\quad {\tau }_{i}\gt {\tau }_{i+1},\\ &&{\tau }_{i}\,{\tau }_{i+1}\,\overset{q}{\to }\,{\tau }_{i+1}\,{\tau }_{i}\quad {\rm{if}}\quad {\tau }_{i}\lt {\tau }_{i+1},\end{eqnarray} \tag{ 2.1 }$$

where q is a free positive parameter. These rules show that particles are ordered by their species: the species N has the highest priority, followed by species $(N-1)$, down to particles of species 2, and lastly by holes (i.e. species 1). Species with higher priority will be said to be faster, so that species N is the fastest species (it is the flash) and species 1 the slowest⁴ .

The bulk rules can be encoded in a local Markov matrix acting on two sites, i.e. on ${{\mathbb{C}}}^{N}\otimes {{\mathbb{C}}}^{N}$. Explicitly, it has the form

$$\begin{eqnarray}&&{\mathfrak{m}}=\displaystyle \sum _{1\leqslant i\lt j\leqslant N}\{({E}_{{ij}}\otimes {E}_{{ji}}-{E}_{{jj}}\otimes {E}_{{ii}})+q({E}_{{ji}}\otimes {E}_{{ij}}-{E}_{{ii}}\otimes {E}_{{jj}})\},\end{eqnarray} \tag{ 2.2 }$$

where E_ij is the N × N elementary matrix with 1 at position (i, j) and 0 elsewhere. This matrix can be obtained from an R-matrix satisfying the Yang–Baxter equation which allows us to prove integrability of the model: this construction will be briefly recalled in the next section. The complete Markov matrix in the bulk is given by

$$\begin{eqnarray}&&{M}_{\mathrm{bulk}}={{\mathfrak{m}}}_{12}+{{\mathfrak{m}}}_{23}\,+\,\cdots \,+\,{{\mathfrak{m}}}_{L-1,L},\end{eqnarray} \tag{ 2.3 }$$

where the indices on ${\mathfrak{m}}$ indicate on which copies of ${{\mathbb{C}}}^{N}$ it acts non-trivially.

On the boundaries. Particles are allowed to enter or to exit from both boundaries and the corresponding entrance/exit rates may depend on the type of the particle that was previously located at the boundary. More precisely, both on the left and on the right boundary, we can have a transition of the type

$$\begin{eqnarray}&&{\tau }_{1}\,\overset{r({\tau }_{1},{\tau }_{2})}{\longrightarrow }\,{\tau }_{2},\end{eqnarray} \tag{ 2.4 }$$

for ${\tau }_{1},{\tau }_{2}=1,2,\ldots ,N$. This leads to $2N(N-1)$ independent rates (that is $N(N-1)$ rates on each side). The rates corresponding to the left boundary are gathered in an N × N boundary matrix B:

$$\begin{eqnarray}&&B=\displaystyle \sum _{1\leqslant i\ne j\leqslant N}r({\tau }_{i},{\tau }_{j}){E}_{{ji}}-\displaystyle \sum _{i=1}^{N}\left(\displaystyle \sum _{j\ne i}r({\tau }_{i},{\tau }_{j})\right){E}_{{ii}}.\end{eqnarray} \tag{ 2.5 }$$

Similarly, the rates for the right boundary are gathered in a matrix $\bar{B}$. An open multi-species ASEP will be defined by the bulk matrix (2.3) and the two boundary matrices B and $\bar{B}$. The Markov matrix associated to the model will be

$$\begin{eqnarray}&&M={M}_{\mathrm{bulk}}+{B}_{1}+{\bar{B}}_{L},\end{eqnarray} \tag{ 2.6 }$$

and the master equation, governing the time evolution of the probability ${P}_{t}({\tau }_{1},\ldots ,{\tau }_{L})$ to be in the configuration $({\tau }_{1},\ldots ,{\tau }_{L})$, is written

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}| {P}_{t}\rangle }{{\rm{d}}t}=M| {P}_{t}\rangle \quad {\rm{where}}\quad | {P}_{t}\rangle =\displaystyle \sum _{1\leqslant {\tau }_{1},\ldots ,{\tau }_{L}\leqslant N}{P}_{t}({\tau }_{1},\ldots ,{\tau }_{L})| {\tau }_{1}\rangle \otimes \cdots \,\otimes \,| {\tau }_{L}\rangle .\end{eqnarray} \tag{ 2.7 }$$

Although the bulk part M_bulk corresponds to an integrable model, for arbitrary choices of the boundary rates, the model will not be integrable. The first result of the present paper is to provide classes of explicit solutions for integrable boundary matrices for the multi-species ASEP. They are presented in section 3.1.

We briefly recall the context of the quantum inverse scattering method that allows one to define open integrable models. We refer to the historical paper [46] and to the review [16] for more details.

We introduce an R-matrix acting on two copies of ${{\mathbb{C}}}^{N}$. It obeys the Yang–Baxter equation and the unitarity relation:

$$\begin{eqnarray}&&{R}_{12}\left(\displaystyle \frac{{x}_{1}}{{x}_{2}}\right){R}_{13}\left(\displaystyle \frac{{x}_{1}}{{x}_{3}}\right){R}_{23}\left(\displaystyle \frac{{x}_{2}}{{x}_{3}}\right)={R}_{23}\left(\displaystyle \frac{{x}_{2}}{{x}_{3}}\right){R}_{13}\left(\displaystyle \frac{{x}_{1}}{{x}_{3}}\right){R}_{12}\left(\displaystyle \frac{{x}_{1}}{{x}_{2}}\right),\end{eqnarray} \tag{ 2.8 }$$

$$\begin{eqnarray}&&{R}_{12}(x){R}_{21}\left(\displaystyle \frac{1}{x}\right)=1.\end{eqnarray} \tag{ 2.9 }$$

Again, the indices indicate in which copies of ${{\mathbb{C}}}^{N}$ the R-matrices act non-trivially. For the multi-species ASEP, the R-matrix can be written as follows

$$\begin{eqnarray}&&{R}_{i,i+1}(x)={P}_{i,i+1}\left(1+\displaystyle \frac{x-1}{{qx}-1}\,{{\mathfrak{m}}}_{i,i+1}\right),\end{eqnarray} \tag{ 2.10 }$$

where P is the permutation operator that exchanges the two copies of ${{\mathbb{C}}}^{N}$ in ${{\mathbb{C}}}^{N}\otimes {{\mathbb{C}}}^{N}$.

To define an integrable open model, one introduces the transfer matrix [46]:

$$\begin{eqnarray}&&{{\mathfrak{t}}}_{\mathrm{open}}(x)={{\tr }}_{0}({R}_{0L}(x)...{R}_{01}(x){K}_{0}(x){R}_{10}(x)...{R}_{L0}(x){\tilde{K}}_{0}(x)),\end{eqnarray} \tag{ 2.11 }$$

where K(x) is a N × N matrix which obeys the reflection equation and is unitary:

$$\begin{eqnarray}&&{R}_{12}\left(\displaystyle \frac{{x}_{1}}{{x}_{2}}\right){K}_{1}({x}_{1}){R}_{21}({x}_{1}{x}_{2}){K}_{2}({x}_{2})={K}_{2}({x}_{2}){R}_{12}({x}_{1}{x}_{2}){K}_{1}({x}_{1}){R}_{21}\left(\displaystyle \frac{{x}_{1}}{{x}_{2}}\right),\end{eqnarray} \tag{ 2.12 }$$

$$\begin{eqnarray}&&K(x)K\left(\displaystyle \frac{1}{x}\right)=1.\end{eqnarray} \tag{ 2.13 }$$

The boundary matrix $\tilde{K}(x)$ in (2.11) satisfies a dual reflection equation. The solutions to this dual reflection equation can be obtained from the solutions K(x) of the reflection equation (2.12) by

$$\begin{eqnarray}&&\widetilde{{K}_{1}}(x)={{\tr }}_{0}({\bar{K}}_{0}(1/x){({({R}_{01}{({x}^{2})}^{{t}_{1}})}^{-1})}^{{t}_{1}}\,{P}_{01}),\end{eqnarray} \tag{ 2.14 }$$

where

$$\begin{eqnarray}\bar{K}(x)={UK}(1/x)U\quad {\rm{and}}\quad U=\left(\begin{array}{ccc} & & 1\\ & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & \\ 1 & & \end{array}\right).\end{eqnarray} \tag{ 2.15 }$$

From these properties, usual calculations [46] prove that the transfer matrix ${{\mathfrak{t}}}_{\mathrm{open}}(x)$ defines an integrable model: $[{{\mathfrak{t}}}_{\mathrm{open}}(x),{{\mathfrak{t}}}_{\mathrm{open}}(y)]=0$. The global Markov matrix is then defined as

$$\begin{eqnarray}&&M=\displaystyle \frac{q-1}{2}{\left.\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}{{\mathfrak{t}}}_{\mathrm{open}}(x)\right|}_{x=1}.\end{eqnarray} \tag{ 2.16 }$$

Then the integrable boundaries are obtained from the K-matrices by

$$\begin{eqnarray}&&B=\displaystyle \frac{q-1}{2}{\left.\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}K(x)\right|}_{x=1}\quad {\rm{and}}\quad \bar{B}=-\displaystyle \frac{q-1}{2}{\left.\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}\bar{K}(x)\right|}_{x=1}.\end{eqnarray} \tag{ 2.17 }$$

We wish to give explicit solutions for integrable Markovian boundary matrices for the multi-species ASEP. These solutions are obtained with relations (2.17) from K-matrices obeying the reflection equation (2.12) with the R-matrix (2.10). We present here the integrable Markovian boundary conditions. We postpone the presentation of the associated K-matrices to section 3.5 and the proof of the integrability to section 4.3.

The integrable boundary conditions depend on two free real positive parameters (rates) α and γ, and four positive integers ${s}_{1},{s}_{2},{f}_{1}$ and f₂, that label two special slow (s) and two special fast (f) species, with the conditions

$$\begin{eqnarray}&&1\leqslant {s}_{1}\leqslant {s}_{2}\lt {f}_{2}\leqslant {f}_{1}\leqslant N\quad {\rm{and}}\quad {f}_{1}-{f}_{2}={s}_{2}-{s}_{1}.\end{eqnarray} \tag{ 3.1 }$$

The four special species will be essentially created on the boundary, while the remaining species will essentially (but not only) decay onto these four types. Any species in between s₁ and s₂ will be paired with one species in between f₂ and f₁, allowing a transmutation (on the boundary) between the pairs. Finally, in between s₂ and f₂, either nothing happens, or the species decay to s₂ and f₂.

More specifically, integrability is preserved when we have the following rules and rates on the boundary:

• Class of very slow species: for species τ with $1\leqslant \tau \lt {s}_{1}$, we have:
  $$\begin{eqnarray}&&\tau \ \overset{\gamma }{\to }\ {s}_{1}\quad {\rm{and}}\quad \tau \ \overset{\alpha }{\to }\ {f}_{1}.\end{eqnarray} \tag{ 3.2 }$$
• Class of slow species: for species τ with ${s}_{1}\leqslant \tau \leqslant {s}_{2}$, we have:
  $$\begin{eqnarray}&&\tau \ \overset{\alpha }{\to }\ \bar{\tau }={s}_{1}+{f}_{1}-\tau ={s}_{2}+{f}_{2}-\tau .\end{eqnarray} \tag{ 3.3 }$$
• Class of intermediate species: for species τ with ${s}_{2}\lt \tau \lt {f}_{2}$, we have the two possibilities:
  • (1)  
    $\tau \ \overset{0}{\to }\ \tau ^{\prime} ,\forall \tau ^{\prime}$ (no decay, creation or transmutation).
  • (2)  
    $\tau \ \overset{\widetilde{\gamma }}{\to }\ {s}_{2}$ and $\tau \ \overset{\widetilde{\alpha }}{\to }\ {f}_{2}$.
• Class of fast species: for species τ with ${f}_{2}\leqslant \tau \leqslant {f}_{1}$, we have:
  $$\begin{eqnarray}&&\tau \ \overset{\widetilde{\gamma }}{\longrightarrow }\ \bar{\tau }={s}_{1}+{f}_{1}-\tau .\end{eqnarray} \tag{ 3.4 }$$
• Class of very fast species: for species τ with ${f}_{1}\lt \tau \leqslant N$, we have:
  $$\begin{eqnarray}&&\tau \ \overset{\widetilde{\gamma }}{\to }\ {s}_{1}\quad {\rm{and}}\quad \tau \ \overset{\widetilde{\alpha }}{\to }\ {f}_{1}.\end{eqnarray} \tag{ 3.5 }$$

We have introduced the following combination of the rates:

$$\begin{eqnarray}&&\widetilde{\alpha }=\displaystyle \frac{(\alpha +\gamma +q-1)\alpha }{\alpha +\gamma },\qquad \widetilde{\gamma }=\displaystyle \frac{(\alpha +\gamma +q-1)\gamma }{\alpha +\gamma }.\end{eqnarray} \tag{ 3.6 }$$

This implies that α, γ, q are constrained such that $\widetilde{\alpha }$, $\widetilde{\gamma }$ are positive.

Note that, depending on the choice of s₁, s₂, f₂ and f₁, some classes of species may not occur: for instance if ${s}_{1}=1$, there is no very slow species. In the same way, if ${f}_{2}={s}_{2}+1$, there are no intermediate species.

Due to the second constraint in (3.1), the number of slow species coincides with the number of fast species, in accordance with the pairing mentioned above. By counting the number of possibilities for s₁, s₂, f₁ and f₂ with the constraints (3.1), we can deduce that, for multi-species ASEP there exist⁵ $\left(\begin{array}{c}N+1\\ 3\end{array}\right)$ different integrable boundaries, each of them depending on two real parameters.

Note that in any transition, the number of particles for the species in the very slow and very fast classes can only decrease. It may stay constant for the slow, fast and intermediate classes. For the four special types it may increase.

To summarize, these rates are gathered in the two following types of boundary matrices, depending on the two possibilities for intermediate species:

$$\begin{eqnarray}&&{B}^{0}(\alpha ,\gamma | {s}_{1},{s}_{2},{f}_{2},{f}_{1})\,=\\ \left(\begin{array}{ccccccccccccccccc}-\sigma & & & & & & & & & & & & & & & & \\ & \ddots & & & & & & & & & & & & & & & \\ & & -\sigma & & & & & & & & & & & & & & \\ \gamma & \cdots & \gamma & -\alpha & & & & & & & & & & \widetilde{\gamma } & \widetilde{\gamma } & \cdots & \widetilde{\gamma }\\ & & & & -\alpha & & & & & & & & \widetilde{\gamma } & & & & \\ & & & & & \ddots & & & & & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & & & & \\ & & & & & & -\alpha & & & & \widetilde{\gamma } & & & & & & \\ & & & & & & & 0 & \cdots & 0 & & & & & & & \\ & & & & & & & \vdots & & \vdots & & & & & & & \\ & & & & & & & 0 & \cdots & 0 & & & & & & & \\ & & & & & & \alpha & & & & -\widetilde{\gamma } & & & & & & \\ & & & & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & & & & & \ddots & & & & & \\ & & & & \alpha & & & & & & & & -\widetilde{\gamma } & & & & \\ \alpha & \cdots & \alpha & \alpha & & & & & & & & & & -\widetilde{\gamma } & \widetilde{\alpha } & \cdots & \widetilde{\alpha }\\ & & & & & & & & & & & & & & -\widetilde{\sigma } & & \\ & & & & & & & & & & & & & & & \ddots & \\ & & & & & & & & & & & & & & & & -\widetilde{\sigma }\end{array}\right),\end{eqnarray} \tag{ 3.7 }$$

$$\begin{eqnarray}&&B(\alpha ,\gamma | {s}_{1},{s}_{2},{f}_{2},{f}_{1})\,=\\ \left(\begin{array}{ccccccccccccccccc}-\sigma & & & & & & & & & & & & & & & & \\ & \ddots & & & & & & & & & & & & & & & \\ & & -\sigma & & & & & & & & & & & & & & \\ \gamma & \cdots & \gamma & -\alpha & & & & & & & & & & \widetilde{\gamma } & \widetilde{\gamma } & \cdots & \widetilde{\gamma }\\ & & & & -\alpha & & & & & & & & \widetilde{\gamma } & & & & \\ & & & & & \ddots & & & & & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & & & & \\ & & & & & & -\alpha & \widetilde{\gamma } & \cdots & \widetilde{\gamma } & \widetilde{\gamma } & & & & & & \\ & & & & & & & -\sigma ^{\prime} & & & & & & & & & \\ & & & & & & & & \ddots & & & & & & & & \\ & & & & & & & & & -\sigma ^{\prime} & & & & & & & \\ & & & & & & \alpha & \alpha & \cdots & \alpha & -\widetilde{\gamma } & & & & & & \\ & & & & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & & & & & \ddots & & & & & \\ & & & & \alpha & & & & & & & & -\widetilde{\gamma } & & & & \\ \alpha & \cdots & \alpha & \alpha & & & & & & & & & & -\widetilde{\gamma } & \widetilde{\alpha } & \cdots & \widetilde{\alpha }\\ & & & & & & & & & & & & & & -\widetilde{\sigma } & & \\ & & & & & & & & & & & & & & & \ddots & \\ & & & & & & & & & & & & & & & & -\widetilde{\sigma }\end{array}\right).\end{eqnarray} \tag{ 3.8 }$$

We have introduced $\sigma =\alpha +\gamma$, $\widetilde{\sigma }=\widetilde{\alpha }+\widetilde{\gamma }$ and $\sigma ^{\prime} =\alpha +\widetilde{\gamma }$. The empty spaces in the matrices above are filled with zeros, and the lines indicate the positions of the four special types of species.

We remark that the rates can be written in a more symmetric way by introducing the combination of parameters⁶

$$\begin{eqnarray}&&a=-\displaystyle \frac{\alpha }{\gamma },\qquad c=\displaystyle \frac{\alpha +\gamma }{\alpha +\gamma +q-1},\end{eqnarray} \tag{ 3.9 }$$

which have previously appeared as the parameters of the Askey–Wilson polynomials in the context of the ASEP stationary state [48]. Then

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{{ac}(1-q)}{(a-1)(c-1)}\qquad \mathrm{and}\qquad \gamma =-\displaystyle \frac{c(1-q)}{(a-1)(c-1)},\end{eqnarray} \tag{ 3.10 }$$

and the parameters $\widetilde{\alpha }$ and $\widetilde{\gamma }$ become

$$\begin{eqnarray}&&\widetilde{\alpha }=\displaystyle \frac{a(1-q)}{(a-1)(c-1)}\qquad \mathrm{and}\qquad \widetilde{\gamma }=-\displaystyle \frac{(1-q)}{(a-1)(c-1)}.\end{eqnarray} \tag{ 3.11 }$$

More solutions: One can produce more integrable solutions using conjugation by any diagonal invertible matrix V. Indeed, due to the invariance of the R-matrix (2.10) by the conjugation by ${V}_{1}{V}_{2}$, ${VK}(x){V}^{-1}$ is solution of the reflection equation if K(x) is also a solution. However, the resulting conjugated matrix may not be Markovian. Nonetheless, we remark that conjugation by the diagonal matrix $\mathrm{diag}({e}^{{s}_{1}},{e}^{{s}_{2}},\ldots ,{e}^{{s}_{N}})$ provides a deformed integrable boundary matrix that allows one to compute the cumulants of the currents at the boundary for the different species.

A right boundary matrix $\bar{B}$ is obtained through the relation (2.17), where $\bar{K}(x)$ is deduced from a solution K(x) thanks to (2.15). Let us stress that the parameters entering $\bar{B}$ are independent from the ones used in the left boundary B. Altogether we will have four real parameters: $\alpha ,\gamma$ for the left boundary, and $\beta ,\delta$ for the right one. In the same way, the labels ${s}_{1}^{\prime },{s}_{2}^{\prime },{f}_{2}^{\prime },{f}_{1}^{\prime }$ of the four special species in the right boundary are independent from the four special species labels ${s}_{1},{s}_{2},{f}_{2},{f}_{1}$ in the left boundary. Explicitly, the right boundary matrices are obtained from relation (2.15) and take the form

$$\begin{eqnarray}&&\bar{B}(\beta ,\delta | {s}_{1}^{\prime },{s}_{2}^{\prime },{f}_{2}^{\prime },{f}_{1}^{\prime })=U\,B(\beta ,\delta | {s}_{1}^{\prime\prime },{s}_{2}^{\prime\prime },{f}_{2}^{\prime\prime },{f}_{1}^{\prime\prime }){U}^{-1},\end{eqnarray} \tag{ 3.12 }$$

where U is defined in (2.15). The conjugation by U implies ${f}_{j}^{\prime\prime }=N+1-{s}_{j}^{\prime }$ and ${s}_{j}^{\prime\prime }=N+1-{f}_{j}^{\prime }$, j = 1, 2.

The bijection between right and left boundaries can be seen in the following identity

$$\begin{eqnarray}&&\bar{B}(\beta ,\delta | {s}_{1},{s}_{2},{f}_{2},{f}_{1})\equiv {B(\delta ,\beta | {s}_{1},{s}_{2},{f}_{2},{f}_{1})| }_{z\leftrightarrow \tilde{z}},\end{eqnarray} \tag{ 3.13 }$$

where $z\leftrightarrow \tilde{z}$ means that we interchange β with $\widetilde{\beta }$ and δ with $\widetilde{\delta }$. As in the case of left boundaries, we use the notation

$$\begin{eqnarray}&&\widetilde{\beta }=\displaystyle \frac{(\beta +\delta +q-1)\beta }{\beta +\delta }\quad {\rm{and}}\quad \widetilde{\delta }=\displaystyle \frac{(\beta +\delta +q-1)\delta }{\beta +\delta }.\end{eqnarray} \tag{ 3.14 }$$

For the case N = 2, we recover the one-species ASEP. We get only one possible choice for s₁, s₂, f₁ and f₂ given by ${s}_{1}={s}_{2}=1$ and ${f}_{1}={f}_{2}=2$. Then, in the language used in this paper, the particle 1 (vacancy) is slow and the particle 2 is fast and the rates at the boundary are given by

$$\begin{eqnarray}&&1\ \overset{\alpha }{\to }\ 2\quad {\rm{and}}\quad 2\ \overset{\widetilde{\gamma }}{\to }\ 1.\end{eqnarray} \tag{ 3.15 }$$

One recovers that for the one-species ASEP, the generic boundary is integrable. The boundary matrix has the form

$$\begin{eqnarray}B=\left(\begin{array}{cc}-\alpha & \widetilde{\gamma }\\ \alpha & -\widetilde{\gamma }\end{array}\right).\end{eqnarray} \tag{ 3.16 }$$

One can use Bethe ansatz method to compute the eigenvalues and compute for example the spectral gap [17].

Conjugation by a diagonal matrix provides the non-Markovian boundary matrix used to compute the cumulant of the current [26]:

$$\begin{eqnarray}B(s)=\left(\begin{array}{cc}-\alpha & {e}^{s}\,\widetilde{\gamma }\\ {e}^{-s}\,\alpha & -\widetilde{\gamma }\end{array}\right).\end{eqnarray} \tag{ 3.17 }$$

It still corresponds to an integrable boundary.

For the case N = 3, we obtain the two-species ASEP. There are four possibilities summarized in table 1. We recover the boundaries found in [15].

Table 1.  The four integrable boundaries in the case N = 3. The last row corresponds to the names of these boundaries in [15].

	${s}_{1}={s}_{2}=1$	${s}_{1}={s}_{2}=2$		${s}_{1}={s}_{2}=1$
	${f}_{1}={f}_{2}=2$	${f}_{1}={f}_{2}=3$	${s}_{1}={s}_{2}=3$
$\begin{array}{c}\mathrm{Type}\ \mathrm{of}\\ \mathrm{Part}.\end{array}$	$\begin{array}{c}\mathrm{Part}.\ 1\ \mathrm{slow}\\ \mathrm{Part}.\ 2\ \mathrm{fast}\\ \mathrm{Part}.\ 3\ \mathrm{very}\ \mathrm{fast}\end{array}$	$\begin{array}{c}\mathrm{Part}.\ 1\ \mathrm{very}\ \mathrm{slow}\\ \mathrm{Part}.\ 2\ \mathrm{slow}\\ \mathrm{Part}.\ 3\ \mathrm{fast}\end{array}$	$\begin{array}{c}\mathrm{Part}.\ 1\ \mathrm{slow}\\ \mathrm{Part}.\ 2\ \mathrm{intermediate}\\ \mathrm{Part}.\ 3\ \mathrm{fast}\end{array}$
	$1\ \overset{\alpha }{\to }\ 2$	$1\ \overset{\gamma }{\to }\ 2$		$1\ \overset{\alpha }{\to }\ 3$
Rates	$2\ \overset{\widetilde{\gamma }}{\to }\ 1$	$1\ \overset{\alpha }{\to }\ 3$	$1\ \overset{\alpha }{\to }\ 3$	$2\ \overset{\widetilde{\gamma }}{\to }\ 1$
	$3\ \overset{\widetilde{\gamma }}{\to }\ 1$	$2\ \overset{\alpha }{\to }\ 3$	$3\ \overset{\widetilde{\gamma }}{\to }\ 1$	$2\ \overset{\alpha }{\to }\ 3$
	$3\ \overset{\widetilde{\alpha }}{\to }\ 2$	$3\ \overset{\widetilde{\gamma }}{\to }\ 2$		$3\ \overset{\widetilde{\gamma }}{\to }\ 1$
Name in [15]	L1	L2	L4	L3

Generic examples. Some of the boundary matrices can be related to former studies of boundary Hecke algebras [22, 33] (see also section 4.1). In our notation, they correspond to the matrices $B(\alpha ,\gamma | 1,{s}_{2},N+1-{s}_{2},N)$ or ${B}^{0}(\alpha ,\gamma | 1,{s}_{2},N+1-{s}_{2},N)$. Among them, some have been considered: ${B}^{0}(\alpha ,\gamma | 1,2,N-1,N)$ was analyzed in [39], and for the two-species ASEP (N = 3) ${B}^{0}(\alpha ,\gamma | 1,1,3,3)$ was studied in [8, 13, 47].

Since the boundary matrices we exhibited depend only on two different rates, one can wonder if, when using these boundaries, the open multi-species ASEP 'trivialises' for N big enough. More precisely, one may ask whether there is some limit on the number of species above which a multi-species ASEP can always be mapped (through identification) to an ASEP with a smaller number of species. In fact, it is not the case, thanks to the four types of special species, that can be chosen freely on each of the two boundaries. Indeed it can be shown that there are pairs of boundary matrices for which the number of particles of any given species is not conserved. Moreover, for any given subset of species, the total number of particles whose species is in this subset is not conserved either.

We give below examples of such pairings of boundary matrices. We write them as $B=B(\alpha ,\gamma | {s}_{1},{s}_{2},{f}_{2},{f}_{1})$ and $\bar{B}=\bar{B}(\beta ,\delta | {s}_{1}^{\prime },{s}_{2}^{\prime },{f}_{2}^{\prime },{f}_{1}^{\prime })$, where the first matrix represents the left boundary, and the second matrix the right one. The explicit values of the four special species (for each boundary) depends on the parity of N in the multi-species ASEP:

For the multi-species ASEP with $N=2n+1$, we can consider the matrices $B=B(\alpha ,\gamma | 2,n+1,n+2,2n+1)$ and $\bar{B}=\bar{B}(\beta ,\delta | 1,n,n+1,2n)$. Explicitly, they are given by

$$\begin{eqnarray*}B=\left(\begin{array}{ccccccc}-\sigma & & & & & & \\ \gamma & -\alpha & & & & & \widetilde{\gamma }\\ & & \ddots & & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & \\ & & & -\alpha & \widetilde{\gamma } & & \\ & & & \alpha & -\widetilde{\gamma } & & \\ & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & & \ddots & \\ \alpha & \alpha & & & & & -\widetilde{\gamma }\end{array}\right)\quad {\rm{with}}\quad \sigma =\alpha +\gamma ,\end{eqnarray*}$$

$$\begin{eqnarray*}\bar{B}=\left(\begin{array}{ccccccc}-\widetilde{\delta } & & & & & \beta & \beta \\ & \ddots & & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & \\ & & -\widetilde{\delta } & \beta & & & \\ & & \widetilde{\delta } & -\beta & & & \\ & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & & \ddots & & \\ \widetilde{\delta } & & & & & -\beta & \delta \\ & & & & & & -\sigma \end{array}\right)\quad {\rm{with}}\quad \sigma =\beta +\delta .\end{eqnarray*}$$

In both cases, the intermediate particles drop out because we choose ${f}_{2}={s}_{2}+1$ and the very fast (respectively very slow) particles do not exist in B (respectively in $\bar{B}$). Then, we have drawn only the line corresponding to s₁ in B and to f₁ in $\bar{B}$.

The evolution of the system given by the Markov chain with these boundaries does not preserve the number of particles of any subset of species. To prove that, we can see that there exists a cycle that connects all the species of the particles and the holes:

$$\begin{eqnarray}1 & \overset{\alpha }{\longrightarrow } & 2n+1 & \overset{\widetilde{\gamma }}{\,\longrightarrow } & 2 & \overset{\widetilde{\delta }}{\longrightarrow \,} & 2n-1 & \overset{\widetilde{\gamma }}{\,\longrightarrow } & 4 & \overset{\widetilde{\delta }}{\longrightarrow \,} & \ldots & \\ \ \uparrow \beta & & & & & & & & & & & \vdots \\ 2n & \overset{\alpha }{\longleftarrow } & 3 & \overset{\beta }{\longleftarrow \,} & 2n-2 & \overset{\alpha }{\,\longleftarrow } & 5 & \overset{\beta }{\longleftarrow \,} & 2n-4 & \overset{\alpha }{\,\longleftarrow \,} & \ldots & .\end{eqnarray} \tag{ 3.18 }$$

For the multi-species ASEP with $N=2n+2$, we can consider the matrices $B=B(\alpha ,\gamma | 2,n+1,n+3,2n+2)$ and $\bar{B}=\bar{B}(\beta ,\delta | 1,n,n+2,2n+1)$, namely

$$\begin{eqnarray*}B=\left(\begin{array}{cccccccc}-\sigma & & & & & & & \\ \gamma & -\alpha & & & & & & \widetilde{\gamma }\\ & & \ddots & & & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & \\ & & & -\alpha & \widetilde{\gamma } & \widetilde{\gamma } & & \\ & & & & -\sigma ^{\prime} & & & \\ & & & \alpha & \alpha & -\widetilde{\gamma } & & \\ & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & & & \ddots & \\ \alpha & \alpha & & & & & & -\widetilde{\gamma }\end{array}\right)\quad {\rm{with}}\quad \left\{\begin{array}{l}\sigma =\alpha +\gamma \\ \sigma ^{\prime} =\alpha +\widetilde{\gamma }\\ \widetilde{\sigma }=\widetilde{\alpha }+\widetilde{\gamma },\end{array}\right.\end{eqnarray*}$$

$$\begin{eqnarray*}\bar{B}=\left(\begin{array}{cccccccc}-\widetilde{\delta } & & & & & & \beta & \beta \\ & \ddots & & & & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & \\ & & -\widetilde{\delta } & \beta & \beta & & & \\ & & & -\sigma ^{\prime} & & & & \\ & & \widetilde{\delta } & \widetilde{\delta } & -\beta & & & \\ & \raise{-3.5pt}\cdot\cdot\raise{3.5pt}\cdot & & & & \ddots & & \\ \widetilde{\delta } & & & & & & -\beta & \delta \\ & & & & & & & -\sigma \end{array}\right)\quad {\rm{with}}\quad \left\{\begin{array}{l}\sigma =\beta +\delta \\ \sigma ^{\prime} =\beta +\widetilde{\delta }.\end{array}\right.\end{eqnarray*}$$

One sees that now the very fast species have been dropped from B and the very slow species from $\bar{B}$.

Again, there exists a cycle (similar to the one above) that connects all the species and the holes, however its form for the intermediate species depends on the parity of n.

To make contact with K-matrices and integrability, we decompose both matrices (3.7) and (3.8) into three pieces, with

$$\begin{eqnarray}{b}_{0}^{+}=\left(\begin{array}{ccccccc}-\gamma & & & & & & \\ & \ddots & & & & & \\ & & -\gamma & & & & \\ \gamma & \cdots & \gamma & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \end{array}\right)\quad {\rm{and}}\quad {b}_{0}^{-}=\left(\begin{array}{ccccccc} & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & \widetilde{\alpha } & \cdots & \widetilde{\alpha }\\ & & & & -\widetilde{\alpha } & & \\ & & & & & \ddots & \\ & & & & & & -\widetilde{\alpha },\end{array}\right),\end{eqnarray} \tag{ 3.19 }$$

where we draw symbolically the lines corresponding to the four special types of particles, to indicate which part of the matrix we picked up in the boundary matrix to construct ${b}_{0}^{\pm }$. Again, the empty spaces are all filled with zeros. The remaining part is ${b}_{0}=B-({b}_{0}^{+}+{b}_{0}^{-})$, where B is either (3.7) or (3.8). Note that the decomposition is done in such a way that each matrix b₀, ${b}_{0}^{\pm }$ is Markovian.

This decomposition of the boundary $B={b}_{0}+{b}_{0}^{+}+{b}_{0}^{-}$ allows the associated K-matrix to be written as

$$\begin{eqnarray}&&K(x)=1+k(x)\left({b}_{0}+x\,{b}_{0}^{+}+\displaystyle \frac{1}{x}\,{b}_{0}^{-}\right),\end{eqnarray} \tag{ 3.20 }$$

$$\begin{eqnarray}&&{\rm{with}}\quad k(x)=\displaystyle \frac{(\alpha +\gamma )({x}^{2}-1)}{(\gamma x+\alpha )((\alpha +\gamma )(x-1)+(q-1)x)}.\end{eqnarray} \tag{ 3.21 }$$

The function k(x) takes a simpler form in the notations (3.10) and (3.11):

$$\begin{eqnarray}&&k(x)=\displaystyle \frac{(1-c)(1-a)}{(q-1)}\displaystyle \frac{({x}^{2}-1)}{(x-a)(x-c)}.\end{eqnarray} \tag{ 3.22 }$$

From (3.20), it is easy to check that

$$\begin{eqnarray}&&B=\displaystyle \frac{q-1}{2}{\left.\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}K(x)\right|}_{x=1}.\end{eqnarray} \tag{ 3.23 }$$

In the next section, we prove the integrability of the K-matrix (3.20) through an algebraic approach.

Before presenting the algebra to construct the boundary, let us recall the construction for the R-matrix based on the Baxterization of the Hecke algebra [28].

For $i=1,2,\ldots ,L-1$, we define the following operators

$$\begin{eqnarray}&&{\check{{ \mathcal R }}}_{i}(x)={{xe}}_{i}-{e}_{i}^{-1}.\end{eqnarray} \tag{ 4.1 }$$

It is well-known that if the generators e_i, $1\leqslant i\leqslant L-1$ obey the so-called Hecke relations, for $i=1,2,\ldots ,L-1$ and $j=1,2,\ldots ,L-2$

$$\begin{eqnarray}&&{e}_{i}^{2}=\omega {e}_{i}+1,\qquad {e}_{j}\,{e}_{j+1}\,{e}_{j}={e}_{j+1}\,{e}_{j}\,{e}_{j+1},\end{eqnarray} \tag{ 4.2 }$$

then the $\check{{ \mathcal R }}(x)$ matrix (4.1) obeys the braided Yang–Baxter equation

$$\begin{eqnarray}&&{\check{{ \mathcal R }}}_{i}({x}_{1}){\check{{ \mathcal R }}}_{i+1}({x}_{1}{x}_{2}){\check{{ \mathcal R }}}_{i}({x}_{2})={\check{{ \mathcal R }}}_{i+1}({x}_{2}){\check{{ \mathcal R }}}_{i}({x}_{1}{x}_{2}){\check{{ \mathcal R }}}_{i+1}({x}_{1}),\end{eqnarray} \tag{ 4.3 }$$

and is unitary, up to normalization. Using relation (4.2), the braided R-matrix can be written as follows

$$\begin{eqnarray}&&{\check{{ \mathcal R }}}_{i}(x)=(x-1){e}_{i}+\omega .\end{eqnarray} \tag{ 4.4 }$$

One can show that the local Markov matrix ${\mathfrak{m}}$ provides a representation of the Hecke algebra:

$$\begin{eqnarray}&&{e}_{i}=({{\mathfrak{m}}}_{i,i+1}+q)/\sqrt{q},\end{eqnarray} \tag{ 4.5 }$$

with $\omega =\sqrt{q}-1/\sqrt{q}$. Then the R-matrix (2.10) is written in terms of (4.1) as

$$\begin{eqnarray}&&{R}_{i}(x)=\displaystyle \frac{1}{x\sqrt{q}-1/\sqrt{q}}{P}_{i,i+1}{\check{{ \mathcal R }}}_{i}(x).\end{eqnarray} \tag{ 4.6 }$$

The extra factor is necessary for unitarity. Then relation (4.3) implies relation (2.8).

To summarize, the idea of the Baxterization (4.1) is to get a solution of the Yang–Baxter equation (i.e. an R-matrix depending on a spectral parameter) from a representation of the Hecke algebra. This idea has been intensively used and generalized to try to classify the solutions of the Yang–Baxter equation [5, 7, 10, 14, 35, 49]. Then, it has been extended to the reflection equation [34] through the boundary Hecke algebra [40]. However, these algebras are not sufficient to include all the boundary matrices we have constructed. Below, we present a slightly more general algebraic structure that encompasses all the boundary matrices we found in the previous section, ensuring integrability of the corresponding models.

We give in the following proposition the Baxterization of the K-matrix associated to a Baxterized R-matrix with Hecke algebra.

Proposition 4.1. Let ${e}_{i}\,(i=1,\ldots ,L-1)$ be the generators of the Hecke algebra satisfying (4.2) and ${\check{{ \mathcal R }}}_{i}(x)$ the associated braided R-matrices (4.1). Let us also define

$$\begin{eqnarray}&&\check{K}(x)=(1-(x-1){e}_{0}){\left(1-\left(\displaystyle \frac{1}{x}-1\right){e}_{0}\right)}^{-1}\end{eqnarray} \tag{ 4.7 }$$

with ${e}_{0}$ a supplementary generator. The inverse in (4.7) is understood as the formal series

$$\begin{eqnarray}&&{\left(1-\left(\displaystyle \frac{1}{x}-1\right){e}_{0}\right)}^{-1}=x{(1-(1-x)({e}_{0}+1))}^{-1}=(y+1)\displaystyle \sum _{n=0}^{\infty }{(-y)}^{n}{({e}_{0}+1)}^{n},\end{eqnarray} \tag{ 4.8 }$$

where $y=x-1$.

Then $\check{K}(x)$ is a solution of the braided reflection equation

$$\begin{eqnarray}&&{\check{{ \mathcal R }}}_{1}({x}_{1}/{x}_{2})\check{K}({x}_{1}){\check{{ \mathcal R }}}_{1}({x}_{1}{x}_{2})\check{K}({x}_{2})=\check{K}({x}_{2}){\check{{ \mathcal R }}}_{1}({x}_{1}{x}_{2})\check{K}({x}_{1}){\check{{ \mathcal R }}}_{1}({x}_{1}/{x}_{2})\end{eqnarray} \tag{ 4.9 }$$

if and only if the supplementary generator ${e}_{0}$ satisfies

$$\begin{eqnarray}&&{e}_{1}\ {e}_{0}\ {e}_{1}\ {e}_{0}-{e}_{0}\ {e}_{1}\ {e}_{0}\ {e}_{1}=\omega (\ {e}_{0}^{2}\ {e}_{1}\ {e}_{0}-{e}_{0}\ {e}_{1}\ {e}_{0}^{2}\ ).\end{eqnarray} \tag{ 4.10 }$$

Moreover the operator $\check{K}(x)$ is unitary:

$$\begin{eqnarray}&&\check{K}(x)\check{K}(1/x)=1.\end{eqnarray} \tag{ 4.11 }$$

Proof. We multiply both sides of the braided reflection equation (4.9) on the left and on the right by

$$\begin{eqnarray}&&\displaystyle \frac{{x}_{2}}{{x}_{1}}\left(1-\left(\displaystyle \frac{1}{{x}_{2}}-1\right){e}_{0}\right)=\displaystyle \frac{1}{{x}_{1}}(1+({x}_{2}-1)({e}_{0}+1))\end{eqnarray} \tag{ 4.12 }$$

and use (4.4), (4.7) to get the following equivalent relation

$$\begin{eqnarray}&&(1+{y}_{2}({e}_{0}+1))\ (({x}_{1}-{x}_{2}){e}_{1}+\omega {x}_{2})\ \displaystyle \frac{1}{{x}_{1}}\check{K}({x}_{1})\ (({x}_{1}{x}_{2}-1){e}_{1}+\omega )\ (1-{y}_{2}{e}_{0})\\ &&\quad =(1-{y}_{2}{e}_{0})\ (({x}_{1}{x}_{2}-1){e}_{1}+\omega )\ \displaystyle \frac{1}{{x}_{1}}\check{K}({x}_{1})\ (({x}_{1}-{x}_{2}){e}_{1}+\omega {x}_{2})\ (1+{y}_{2}({e}_{0}+1)),\end{eqnarray} \tag{ 4.13 }$$

where ${y}_{i}={x}_{i}-1$. Then, we use the expansion (4.8) of $\tfrac{1}{{x}_{1}}\check{K}({x}_{1})$ in terms of y₁. The coefficient of ${y}_{1}{y}_{2}^{3}$ in (4.13) provides relation (4.10), which proves that (4.9) implies (4.10).

To prove the reverse implication, we use the following lemma:

Lemma 4.2. Relation (4.10) implies, for $k=0,1,2,\ldots$,

$$\begin{eqnarray}&&{e}_{1}\ {e}_{0}\ {e}_{1}\ {e}_{0}^{k}-{e}_{0}^{k}\ {e}_{1}\ {e}_{0}\ {e}_{1}=\omega ({e}_{0}^{k+1}\ {e}_{1}\ {e}_{0}-{e}_{0}\ {e}_{1}\ {e}_{0}^{k+1}),\end{eqnarray} \tag{ 4.14 }$$

$$\begin{eqnarray}{e}_{1}\ {e}_{0}^{k}\ {e}_{1}\ {e}_{0}-{e}_{0}\ {e}_{1}\ {e}_{0}^{k}\ {e}_{1} & = & \omega ({e}_{0}^{k+1}\ {e}_{1}\ {e}_{0}-{e}_{0}\ {e}_{1}\ {e}_{0}^{k+1}\ \\ & & +{e}_{0}^{k}\ {e}_{1}\ {e}_{0}-{e}_{0}\ {e}_{1}\ {e}_{0}^{k}\ ),\end{eqnarray} \tag{ 4.15 }$$

$$\begin{eqnarray}{e}_{1}\ {({e}_{0}+1)}^{k}\ {e}_{1}\ {e}_{0}-{e}_{0}\ {e}_{1}\ {({e}_{0}+1)}^{k}\ {e}_{1} & = & \omega ({({e}_{0}+1)}^{k+1}\ {e}_{1}\ {e}_{0}\ \\ & & -{e}_{0}\ {e}_{1}\ {({e}_{0}+1)}^{k+1}),\end{eqnarray} \tag{ 4.16 }$$

$$\begin{eqnarray}{e}_{1}\ {e}_{0}{({e}_{0}+1)}^{k}\ {e}_{1}\ {e}_{0}-{e}_{0}\ {e}_{1}\ {e}_{0}{({e}_{0}+1)}^{k}\ {e}_{1} & = & \omega ({e}_{0}\ {({e}_{0}+1)}^{k+1}\ {e}_{1}\ {e}_{0}\ \\ & & -{e}_{0}\ {e}_{1}\ {e}_{0}\ {({e}_{0}+1)}^{k+1}).\end{eqnarray} \tag{ 4.17 }$$

The first relation of the lemma (4.14) is proven by recursion using (4.10). Relation (4.15) is proven also by recursion with (4.14) and (4.2). The third and the fourth are proven by expanding ${({e}_{0}+1)}^{k}$ and using (4.15).

The lemma allows us to prove that

$$\begin{eqnarray}&&{e}_{1}\check{K}(x){e}_{1}{e}_{0}-{e}_{0}{e}_{1}\check{K}({x}_{1}){e}_{1}=\omega (({e}_{0}+1)\check{K}(x){e}_{1}{e}_{0}-{e}_{0}{e}_{1}({e}_{0}+1)\check{K}(x)).\end{eqnarray} \tag{ 4.18 }$$

Finally, by expanding (4.13) and by using relation (4.18), we prove that equation (4.10) implies (4.9) which concludes the proof of the proposition. ■

Connection with Baxterization of cyclotomic Hecke algebras. Another Baxterization for the K-matrix was proposed in [31], starting from a slightly different algebra. There, the relation (4.10) is replaced by

$$\begin{eqnarray}&&{e}_{1}\,{\bar{e}}_{0}\,{e}_{1}\,{\bar{e}}_{0}-{\bar{e}}_{0}\,{e}_{1}\,{\bar{e}}_{0}\,{e}_{1}=0,\end{eqnarray} \tag{ 4.19 }$$

$$\begin{eqnarray}&&\displaystyle \sum _{k=0}^{m}{a}_{k}\,{({\bar{e}}_{0})}^{k}=0\end{eqnarray} \tag{ 4.20 }$$

for some fixed $m=2,3,\ldots$ and a₀, ... a_m free parameters. The relation (4.20) is called the cyclotomic relation, for an introduction to cyclotomic algebras, see e.g. [2]. Then, a K-matrix can be constructed as a polynomial in ${\bar{e}}_{0}$ [31]. When m = 2, the cyclotomic Hecke algebra is just the boundary Hecke algebra.

In fact, similarly to proposition 4.1, one can show that

$$\begin{eqnarray}&&\check{K}(x)=(1-x{\bar{e}}_{0}){\left(1-\displaystyle \frac{1}{x}{\bar{e}}_{0}\right)}^{-1}\end{eqnarray} \tag{ 4.21 }$$

satisfies the reflection equation, provided ${\bar{e}}_{0}$ satisfies solely the relation (4.19). The polynomial Baxterization of [31] is recovered when one assumes in addition the cyclotomic relation (4.20).

One can match this Baxterization with the one presented in (4.7) in the following way. Starting from the algebra (4.10), and assuming that $({e}_{0}+1)$ is invertible, it is possible to prove that the generator

$$\begin{eqnarray}&&{\bar{e}}_{0}={e}_{0}{(1+{e}_{0})}^{-1}\end{eqnarray} \tag{ 4.22 }$$

satisfies the relation (4.19). This can be shown by using relation (4.10) for e₀ and lemma 4.2. Then, substituting (4.22) into the Baxterized K-matrix (4.21) yields (4.7) up to a normalization factor.

The aim of this section is to prove that the boundary matrices presented in section 3.1 fit into the Baxterization procedure of proposition 4.1.

Proposition 4.3. For any matrix $B=B(\alpha ,\beta | {s}_{1},{s}_{2},{f}_{2},{f}_{1})$ or $B={B}^{0}(\alpha ,\beta | {s}_{1},{s}_{2},{f}_{2},{f}_{1})$, the generators

$$\begin{eqnarray}&&{e}_{0}=\displaystyle \frac{B+\alpha +\gamma +q-1}{1-q}\,{and}\,{e}_{1}=({\mathfrak{m}}+q)/\sqrt{q}\end{eqnarray} \tag{ 4.23 }$$

obey relation (4.10), where ${\mathfrak{m}}\equiv {{\mathfrak{m}}}_{12}$ is given in (2.2) and $B$ acts non trivially in space $1$.

Proof. The matrices e₁ and e₀ given in (4.5) and (4.23) act on two site multi-species ASEP configurations. For a given start state, ${\tau }_{1}{\tau }_{2}$, we can find a subset of the particle species ${ \mathcal S }=\{{\tau }_{1},{\tau }_{2},{\tau }_{3},\ldots \}$ such that for any polynomial in e₁ and e₀ acting on this state, these are the only species involved in the resulting configurations.

For all of the boundary matrices we consider, the subset ${ \mathcal S }$ turns out to be small, and related to the different classes of particles we introduced above: the non-diagonal part of e₁ exchanges particles on sites 1 and 2, as allowed by bulk matrix ${\mathfrak{m}};$ the non-diagonal part of e₀ injects and removes particles at site 1 as allowed by the boundary transitions given in section 3.1. The idea of the proof is then to project the 'global' matrices e₀, e₁ down to the smaller number of species in ${ \mathcal S }$. If for every starting state we can show that the resulting projected e₀, e₁ satisfy (4.10), then this implies that the 'global' matrices also satisfy (4.10).

At this point, the proof decomposes into different steps:

• We remark that for any start state ${\tau }_{1}{\tau }_{2}$, the set ${ \mathcal S }$ falls into one of three categories:
  $$\begin{eqnarray}&&{ \mathcal S }=\{{\tau }_{1},{\tau }_{2},{s}_{1},{s}_{2},{f}_{1},{f}_{2}\},\end{eqnarray} \tag{ 4.24 }$$
  $$\begin{eqnarray}&&{ \mathcal S }=\{{\tau }_{1},{s}_{1}+{f}_{1}-{\tau }_{1},{\tau }_{2},{s}_{1}+{f}_{1}-{\tau }_{2}\},\end{eqnarray} \tag{ 4.25 }$$
  $$\begin{eqnarray}&&{ \mathcal S }=\{{\tau }_{1},{s}_{1}+{f}_{1}-{\tau }_{1},{\tau }_{2},s,f\},\quad {\rm{with}}\quad (s,f)=({s}_{1},{f}_{1}){\rm{}}\,{\rm{or}}\,{\rm{}}({s}_{2},{f}_{2}).\end{eqnarray} \tag{ 4.26 }$$
  Note that these sets can be reduced depending on the class of the species ${\tau }_{1}$ and ${\tau }_{2}$. For instance, if ${\tau }_{1}$ and ${\tau }_{2}$ are of very slow class, then ${ \mathcal S }=\{{\tau }_{1},{\tau }_{2},{s}_{1},{f}_{1}\}$. Note also that the ordering of the start state does not change ${ \mathcal S }$ so ${\tau }_{1}$, ${\tau }_{2}$ are interchangeable in (4.26).
• Projecting the boundary matrix, B, corresponding to e₀ down to the species in ${ \mathcal S }$ results in a boundary matrix of size $| { \mathcal S }|$ of type (3.7) or (3.8). To see this, we perform the projection by 'deleting' species from B by removing the corresponding row and column: we use the following operations which preserve the forms (3.7) or (3.8):
  • Deleting any species in the very slow, intermediate, or very fast class;
  • Deleting a species, τ, in the slow or fast class with $\tau \ne {s}_{1},{f}_{1}$ if we also delete the species ${s}_{1}+{f}_{1}-\tau ;$
  • Deleting species s₁ and f₁ together, if ${s}_{1}=1$, ${f}_{1}=N$, and ${f}_{1}-{f}_{2}={s}_{2}-{s}_{1}\gt 0$.
  • Deleting species s₂ and f₂ together, if ${f}_{2}={s}_{2}+1$, and ${f}_{1}-{f}_{2}={s}_{2}-{s}_{1}\gt 0$.
  These operations are always sufficient to project down to any subsets ${ \mathcal S }$ as defined above. The projected e₀ is then obtained from the projected B through (4.23).
• For the local bulk matrix ${\mathfrak{m}}$ (giving e₁) we can delete any number of species, preserving the form (2.2).
• To complete the proof all we need to do is to verify that all boundary matrices in this family give e₀ matrices which satisfy (4.10) for size 2 up to 6 (the maximum $| { \mathcal S }|$). We have done this by a direct computation with a formal mathematical software package.

To illustrate the projection on ${ \mathcal S }$, we consider the following boundary matrix

$$\begin{eqnarray}B=\left(\begin{array}{ccccccc}-\sigma & & & & & & \\ \gamma & -\alpha & & & & \widetilde{\gamma } & \widetilde{\gamma }\\ & & -\alpha & \widetilde{\gamma } & \widetilde{\gamma } & & \\ & & & -\sigma ^{\prime} & & & \\ & & \alpha & \alpha & -\widetilde{\gamma } & & \\ \alpha & \alpha & & & & -\widetilde{\gamma } & \widetilde{\alpha }\\ & & & & & & -\widetilde{\sigma }\end{array}\right)\end{eqnarray} \tag{ 4.27 }$$

and give some examples of start state $({\tau }_{1},{\tau }_{2})$ and the resulting subset ${ \mathcal S }$ and corresponding reduced matrix. In the case where $({\tau }_{1},{\tau }_{2})=(1,4)$, we obtain ${ \mathcal S }=\{1,4,{s}_{1}=2,{s}_{2}=3,{f}_{1}=6,{f}_{2}=5\}$ and the reduced matrix reads

$$\begin{eqnarray}\left(\begin{array}{cccccc}-\sigma & & & & & \\ \gamma & -\alpha & & & & \widetilde{\gamma }\\ & & -\alpha & \widetilde{\gamma } & \widetilde{\gamma } & \\ & & & -\sigma ^{\prime} & & \\ & & \alpha & \alpha & -\widetilde{\gamma } & \\ \alpha & \alpha & & & & -\widetilde{\gamma }\end{array}\right).\end{eqnarray} \tag{ 4.28 }$$

In the case where $({\tau }_{1},{\tau }_{2})=(2,6)$, we obtain ${ \mathcal S }=\{2,6\}$ and the reduced matrix reads

$$\begin{eqnarray}\left(\begin{array}{cc}-\alpha & \widetilde{\gamma }\\ \alpha & -\widetilde{\gamma }\end{array}\right).\end{eqnarray} \tag{ 4.29 }$$

Finally, in the case where $({\tau }_{1},{\tau }_{2})=(3,4)$, we obtain ${ \mathcal S }=\{3,4,5\}$ and the reduced matrix reads

$$\begin{eqnarray}\left(\begin{array}{ccc}-\alpha & \widetilde{\gamma } & \widetilde{\gamma }\\ 0 & -\sigma ^{\prime} & 0\\ \alpha & \alpha & -\widetilde{\gamma }\end{array}\right).\end{eqnarray} \tag{ 4.30 }$$

□

From proposition 4.1, the generator e₀ defined above provides a $\check{K}(x)$ matrix that obeys the reflection equation and is unitary. The connection with the expression (3.20) is given by the following proposition.

Proposition 4.4. For any matrix $B=B(\alpha ,\beta | {s}_{1},{s}_{2},{f}_{2},{f}_{1})$ or $B={B}^{0}(\alpha ,\beta | {s}_{1},{s}_{2},{f}_{2},{f}_{1})$, let $B={b}_{0}+{b}_{0}^{+}+{b}_{0}^{-}$ be the decomposition described in section 3.5. The matrix

$$\begin{eqnarray}&&K(x)=1+k(x)\left({b}_{0}+x\,{b}_{0}^{+}+\displaystyle \frac{1}{x}\,{b}_{0}^{-}\right),\end{eqnarray} \tag{ 4.31 }$$

$$\begin{eqnarray}&&{\rm{with}}\quad k(x)=\displaystyle \frac{({x}^{2}-1)(\alpha +\gamma )}{(\gamma x+\alpha )((\alpha +\gamma )(x-1)+(q-1)x)}\,\end{eqnarray} \tag{ 4.32 }$$

can be expressed as a Baxterized $\check{K}(x)$ matrix

$$\begin{eqnarray}&&K(x)=\displaystyle \frac{(\alpha +\gamma +q-1)\left(\tfrac{1}{x}-1\right)+q-1}{(\alpha +\gamma +q-1)(x-1)+q-1}\left(\displaystyle \frac{1-(x-1){e}_{0}}{1-\left(\tfrac{1}{x}-1\right){e}_{0}}\right),\end{eqnarray} \tag{ 4.33 }$$

where

$$\begin{eqnarray}&&{e}_{0}=\displaystyle \frac{B+\alpha +\gamma +q-1}{1-q}.\end{eqnarray} \tag{ 4.34 }$$

Thus, it satisfies the reflection equation and is unitary.

Proof. Note that K(x) in (4.33) has the form

$$\begin{eqnarray*}&&K(x)=\displaystyle \frac{f(x)}{f(1/x)}\check{K}(x)\end{eqnarray*}$$

with $\check{K}(x)$ as in (4.7) so that it remains unitary and satisfies the reflection equation.

To show that the K-matrix (4.33) is equivalent to the form (4.31) we will need the following relations for the matrices b₀, ${b}_{0}^{+}$ and ${b}_{0}^{-}$:

$$\begin{eqnarray}&&{b}_{0}^{2}=-(\alpha +\widetilde{\gamma }){b}_{0}+\widetilde{\alpha }{b}_{0}^{+}+\gamma {b}_{0}^{-},\qquad {({b}_{0}^{+})}^{2}=-\gamma {b}_{0}^{+},\qquad {({b}_{0}^{-})}^{2}=-\widetilde{\alpha }{b}_{0}^{-},\\ &&{b}_{0}{b}_{0}^{+}={b}_{0}^{+}{b}_{0}=-\alpha {b}_{0}^{+},\qquad {b}_{0}{b}_{0}^{-}={b}_{0}^{-}{b}_{0}=-\widetilde{\gamma }{b}_{0}^{-},\\ &&{b}_{0}^{+}{b}_{0}^{-}={b}_{0}^{-}{b}_{0}^{+}=0.\end{eqnarray} \tag{ 4.35 }$$

They involve the combination of parameters defined in (3.6). These relations are proven using the same method of projecting down to a set ${ \mathcal S }$ of all species involved from a given starting state. In this case, the matrices act on a single site configuration, and it is sufficient to check that the relations hold for size 2 and 3.

Using (4.35) it is straightforward to check that

$$\begin{eqnarray}&&\left(1-\left(\displaystyle \frac{1}{x}-1\right){e}_{0}\right)\left(q-1+\displaystyle \frac{(x-1)(\alpha +\gamma )}{(\alpha +\gamma x)}\left(\displaystyle \frac{{b}_{0}^{-}}{x}+{b}_{0}+{b}_{0}^{+}x\right)\right),\end{eqnarray} \tag{ 4.36 }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{1}{x}(\alpha +\gamma +q-1-(\alpha +\gamma )x),\end{eqnarray} \tag{ 4.37 }$$

which allows ${\left(1-\left(\tfrac{1}{x}-1\right){e}_{0}\right)}^{-1}$ to be expressed as a polynomial in ${b}_{0},{b}_{0}^{\pm }$. Then using (4.35) again we can show that

$$\begin{eqnarray}&&\displaystyle \frac{(\alpha +\gamma +q-1)\left(\tfrac{1}{x}-1\right)+q-1}{(\alpha +\gamma +q-1)(x-1)+q-1}(1-(x-1){e}_{0}){\left(1-\left(\displaystyle \frac{1}{x}-1\right){e}_{0}\right)}^{-1},\end{eqnarray} \tag{ 4.38 }$$

$$\begin{eqnarray}&&=\,1+\displaystyle \frac{({x}^{2}-1)(\alpha +\gamma )}{(\gamma x+\alpha )((\alpha +\gamma )(x-1)+(q-1)x)}\left({b}_{0}+x\,{b}_{0}^{+}+\displaystyle \frac{1}{x}\,{b}_{0}^{-}\right),\end{eqnarray} \tag{ 4.39 }$$

which concludes the proof.□

Polynomial relations. Using relations (4.35) and the expression (4.23) for e₀, we get

$$\begin{eqnarray}&&{e}_{0}({e}_{0}+1)\left({e}_{0}+\displaystyle \frac{\alpha }{\alpha +\gamma }\right)\left({e}_{0}+\displaystyle \frac{\alpha +\gamma +q-1}{q-1}\right)=0.\end{eqnarray} \tag{ 4.40 }$$

We stress that the factor $({e}_{0}+1)$ is present in (4.40). Moreover, we find that for particular choices of b₀, ${b}_{0}^{+}$ and ${b}_{0}^{-}$, the polynomial (4.40) becomes minimal for e₀. Then, in these cases, $({e}_{0}+1)$ is not invertible and we cannot use (4.22) to Baxterize the K-matrix for the multi-species ASEP from the construction of [31]. However, the model is still integrable thanks to the Baxterization (4.7).

Note that when ${b}_{0}^{+}={b}_{0}^{-}=0$, relations (4.35) reduce to

$$\begin{eqnarray}&&{b}_{0}({b}_{0}+\alpha +\widetilde{\gamma })=0\quad {\rm{i.e.}}\quad \left({e}_{0}+\displaystyle \frac{\alpha }{\alpha +\gamma }\right)\left({e}_{0}+\displaystyle \frac{\alpha +\gamma +q-1}{q-1}\right)=0\end{eqnarray} \tag{ 4.41 }$$

and we recover the boundary Hecke algebra.