Weakly involutive $R$-matrices and representations of the symmetric group $S_N$ in restricted subspaces of $V^{\otimes N}$

Question

An $R$-matrix is a matrix $R\in\operatorname{End}(V\otimes V)$ (where $V$ is a finite dimensional vector space) that solves the Yang–Baxter equation $$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$ where both sides are regarded as elements of $\mathrm{End}(V\otimes V\otimes V)$, and $R_{12}=R\otimes I$, $R_{23}=I\otimes R$, $I$ is the identity matrix in $\operatorname{End}(V)$.

We say that an $R$-matrix is involutive if it in addition satisfies $R^2=I\otimes I$. An involutive $R$-matrix defines a representation of the symmetric group $S_N$ in the tensor product vector space $V^{\otimes N}$, where the permutation $(j,j+1)\in S_N$ acts as $I^{\otimes (j-1)}\otimes R\otimes I^{\otimes (N-j-1)}$.

Here comes a new definition: we say that an $R$-matrix is weakly-involutive, if $R^2=P$, where $P\in\operatorname{End}(V\otimes V)$ is a projection operator satisfying $$P^2=P,\text R_{12}P_{23}=P_{23}R_{12}, \text{ and } P_{12}R_{23}=R_{23}P_{12},$$ where, as before, $P_{12}=P\otimes I$, $P_{23}=I\otimes P$.

It is straightforward to verify that a weakly involutive $R$-matrix defines a representation of $S_N$ in the restricted subspace $\mathrm{P}_N\subseteq V^{\otimes N}$, defined by the projection operator $$\hat{\mathrm{P}}_N=\prod_{j=1}^{N-1}P_{j,j+1}.$$ The subspace $\mathrm{P}_{N}$ is spanned by all vectors ${\psi} \in V^{\otimes N}$ satisfying $\hat{\mathrm{P}}_N{\psi}={\psi}$. This definition of weakly-involutive $R$-matrix is motivated by the relaxation of the axioms of quasitriangular Hopf algebras to quasitriangular weak Hopf algebras [Definition 5.1 in Nikshych-Turaev-Vainerman (2000), see also this question].

Furthermore, we define the effective quantum dimension $D$ of a weakly-involutive $R$-matrix as $$D=\lim_{N\to\infty} (\dim \mathrm{P}_N)^{1/N}.$$ Note that we always have $D\leq \dim(V)$, where equality holds if $R$ is involutive.

Question: Are weakly involutive $R$-matrices as defined above studied in the literature? Do there exist non-trivial examples of weakly involutive R-matrices? [Non-trivial means cannot be reduced to an ordinary involutive $R$-matrix in any straightforward way; some minimal requirements are that $0<\operatorname{rank}(P)<(\dim V)^2$, and $P$ is not a product operator, i.e. $P$ is not of the form $P_1\otimes P_2$, where $P_1,P_2\in\operatorname{End}(V)$. I think if the effective quantum dimension $D$ is not an integer, that is a strong evidence of non-triviality.]

Answer 1

In the recent physics preprint

Corcoran, De Leeuw and Pozsgay, Integrable models on Rydberg atom chains [arXiv:2405.15848]

the authors study quantum-integrable models related to $R$-matrices with spectral parameter obeying a condition that seems to be your weakly involutive property. Presumably your setting arises in a 'braid limit' ($u\to\infty$ or so).

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To help understand their notation, here's what I gather from a quick look at the paper.

Consider an $R$-matrix $R(u,v)$ obeying the YBE with spectral parameter (12) in the paper. The authors write $\mathcal{P}$ for the flip $\mathcal{P} \, v\otimes w = w\otimes v$ on $V\otimes V$. Hence the product $\check{R}(u,v) := \mathcal{P} \, R(u,v)$ obeys the YBE in the form $$\check{R}_{12}(u,v) \, \check{R}_{23}(u,w) \, \check{R}_{12}(v,w) = \check{R}_{23}(v,w) \, \check{R}_{12}(u,w) \, \check{R}_{23}(u,v) \, ,$$ which is the generalisation with spectral parameter of the involutive case of what you denote by $R$.

Next, the authors consider a projection operator $\Pi$ (your $P$), and define $\tilde{R}(u,v) := \Pi \, R(u,v)$. It still obeys the YBE with spectral parameter (64), but appears to be only weakly involutive, see (65). More precisely, $\mathcal{P} \, \tilde{R}(u,v)$ obeys $$\mathcal{P} \, \tilde{R}(u,v) \ \mathcal{P} \, \tilde{R}(v,u) = \Pi \, ,$$ which is the natural generalisation with spectral parameter of your weak involutivity property. Note I haven't verified if they require exactly the same conditions as the OP.

If so, this construction seems to provide a source of examples: start from an involutive $R$-matrix, and compose it with some projection operator to get a weakly involutive $R$-matrix.

A specific example that the authors have in mind, see (34)–(35), is $V=\mathbb{C}^2$ with standard basis denoted (in standard quantum mechanics 'ket' notation) by $\lvert \uparrow\rangle, \lvert \downarrow\rangle$. $\Pi$ is the rank-3 projection on $V\otimes V$ with kernel spanned by $\lvert \downarrow\rangle \otimes \lvert \downarrow\rangle$. (It is the identity minus the 'product operator' equal to the rank-1 projection onto the span of $\lvert \downarrow\rangle \otimes \lvert \downarrow\rangle$, so perhaps trivial from your viewpoint?)

The authors go on to show that their setting is enough to apply the usual yoga of quantum-integrable models and construct a Bethe subalgebra, i.e. a family of commuting operators, generated by a 'transfer matrix'. They construct several examples (which perhaps provide further examples of nontrivial weakly involutive $R$-matrices with spectral parameter?) and study some of the resulting models.

PS. I'm on holiday, it's after midnight, and I'm writing on my phone, so apologies for typos and other errors. I hope this is nevertheless of some help.