| Russian Mathematical Surveys |
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Brief communications Universal matrix Capelli identity M. R. Zaitsev National Research University Higher School of Economics, Moscow, Russia
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    1.In this note we present the universal matrix Capelli identity in the reflection equation algebra $M(R)$ associated with an arbitrary Hecke $R$-matrix. Identity (1) involves the matrices $M$ and $D$ , which are quantum analogues of the matrix of variables and the matrix of derivatives, respectively. The matrix $ \widehat L=MD$ is a quantum analogue of the matrix of generators of the universal enveloping algebra $U({\mathfrak{gl}}_{(M|N)})$. The elements $J_i$ denote the $R$-matrix images of Jucys–Murphy elements of the Hecke algebra $\mathbb{H}_n(q)$. 2.For any matrix $A \in \operatorname{Mat}_{N \times N}(\mathbb{C}) \otimes U$ with coefficients in an arbitrary vector space $U$ we use the notation
$$
\begin{equation*}
A_i=\operatorname{Id}_{\mathbb{C}^N} \otimes\operatorname{Id}_{\mathbb{C}^N} \otimes \cdots\otimes\underset{i}{A}\otimes \cdots\otimes \operatorname{Id}_{\mathbb{C}^N}\in \operatorname{Mat}_{N \times N}(\mathbb{C})^{\otimes k}\otimes U.
\end{equation*}
\notag
$$
The reflection equation algebra $M(R)$ is the unital associative algebra generated by $N^2$ generators $m_i^j$, which we combine into a matrix $M=\|m_i^j\|_{1\leqslant i,j \leqslant N}$ subject to the reflection equation:
$$
\begin{equation}
R_{12}M_{\bar{1}} M_{\bar{2}}= M_{\bar{1}} M_{\bar{2}}R_{12}; \quad M_{\bar{1}}=M_1\quad\text{and}\quad M_{\bar{k}}=R_{k-1\,k}M_{\overline{k-1}}R_{k-1\,k}, \quad k>1.
\end{equation}
\tag{2}
$$
Here $R \in \operatorname{Mat}_{N \times N}(\mathbb{C})^{\otimes 2}$ is the Hecke $R$-matrix, that is, $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$, $R^2=1+(q-q^{-1}) R$, $q \in \mathbb{C}\setminus \{0,\pm 1\}$. The parameter $q$ is generic: $q^{k} \ne 1$ for any $k \in \mathbb{Z}_{>0}$. Introducing the new matrix $\widehat L=(1-M)/(q-q^{-1})$ into (2) we obtain
$$
\begin{equation}
\widehat L_1R_{12}\widehat L_1R_{12}-R_{12}\widehat L_{1}R_{12}\widehat L_{1}= \widehat L_{1} R_{12}-R_{12} \widehat L_{1}.
\end{equation}
\tag{3}
$$
The algebra $M(R)$ is embedded into the quantum Weyl algebra $W(R)$, which is the quotient of the free product of two reflection equation algebras, $M(R)$ and $D(R^{-1})$ (in the definition of the latter algebra, $R$ is replaced by $R^{-1}$), modulo the relation $D_1 M_{\bar{2}}= R_{12}^{-1}+M_{\bar{2}}D_1 R_{12}^{-2}$. It is known (see [1]) that the matrix $\widehat L=MD$ satisfies the relation (3), and thus we have a representation of the reflection equation algebra (3) by quantum analogues of vector fields. In this setting, the Jucys–Murphy elements entering (1) are defined by $J_1=1$ and $J_k=R_{k-1\,k} J_{k-1} R_{k-1\,k}$, $k >1$.
3.Let $L=\|L_i^j\|_{1 \leqslant i,j \leqslant N}$ be the matrix composed of the generators of the algebra $U({\mathfrak{gl}}_{N})$, and let $P_{12}$ be the permutation matrix with entries $P_{kl}^{mn}=\delta_k^n\, \delta_l^m$, $ 1\leqslant k,l,m,n\leqslant N$. In terms of the matrix $L$, the relations in the universal enveloping algebra $U({\mathfrak{gl}}_N)$ can be written in the matrix form: $L_1 L_2-L_2 L_1= L_1 P_{12}-P_{12} L_1$. In particular, for the Drinfeld–Jimbo $R$-matrix the algebra defined by (3) is a deformation of the universal enveloping algebra $U({\mathfrak{gl}}_N)$. Moreover, in the case when the $R$-matrix is a deformation of the super-permutation $P_{M|N}$, the algebra $M(R)$ is a deformation of $U({\mathfrak{gl}}_{(M|N)})$. Consider the Weyl algebra $W_N$ generated by the entries of the matrices $X$ and $D$, where $X=\|x_i^j\|_{1 \leqslant i,j \leqslant N}$ and $D = \|\partial_i^j\|_{1 \leqslant i,j \leqslant N}$, $\partial_i^j = \frac{\partial}{\partial x_j^i}$. The relations in $W_N$ are $X_1 X_2=X_2 X_1$, $D_1 D_2=D_2 D_1$, and $D_1 X_2=X_2 D_1+P_{12}$. The algebra $U({\mathfrak{gl}}_N)$ is embedded into the Weyl algebra by the homomorphism that takes elements of $L$ to the corresponding elements of the matrix $XD$. The universal (not quantum) matrix Capelli identity in terms of $L=XD$ takes the form
$$
\begin{equation*}
L_{1}(L_{2}-j_2) \cdots (L_{n}-j_n)=X_{1} \cdots X_{n} D_{1} \cdots D_{n},
\end{equation*}
\notag
$$
where the $j_k=\sum_{i=1}^{k-1} P_{i k}$ represent the images of Jucys–Murphy elements of the group algebra of the symmetric group. As an immediate consequence, we derive Capelli’s identity for quantum immanants (introduced by Okounkov in [2]):
$$
\begin{equation}
\operatorname{Tr}_{(1 \dots n)}\bigl(L_1 (L_2-c(2)) \cdots (L_n-c(n)) P_{ii}^{\lambda}\bigr)=\operatorname{Tr}_{(1 \dots n)} \bigl(X_1 \cdots X_n D_1 \cdots D_n P_{ii}^{\lambda}\bigr).
\end{equation}
\tag{4}
$$
Here $P_{ii}^{\lambda}$ is a primitive idempotent of the group algebra of the symmetric group that corresponds to the standard Young tableau of the form $\lambda$ with index $i$ in any numbering, and $c(k)$ denotes the content of the box with index $k$ in this standard tableau. The symbol $\operatorname{Tr}_{(k)}$ represents the partial trace in the $k$th tensor component. Note that both sides of (4) are independent of the index $i$ of the primitive idempotent $P_{ii}^{\lambda}$. A trace-free version of this identity was proposed in [3]. We call the version from [3], as well as all versions in which traces are not used, matrix versions.
4.We emphasize that for a generic parameter $q$ the Hecke algebra $\mathbb{H}_n(q)$ of type $A_{n-1}$ is semisimple and isomorphic to the group algebra of the symmetric group. In particular, primitive idempotents $E_{ii}^{\lambda}$ of $\mathbb{H}_n(q)$ are in a 1-to-1 correspondence with the standard Young tableaux. The Jucys–Murphy elements act on them by the formula $J_k E_{ii}^{\lambda}=q^{2c(k)} E_{ii}^{\lambda}$. Multiplying (1) by such an idempotent and using the last relation, we obtain the $q$-deformed identity
$$
\begin{equation}
\begin{aligned} \, &\widehat L_{\bar{1}} (\widehat L_{\bar{2}}-q^{-c(2)}[c(2)]_{q})\cdots (\widehat L_{\bar{n}}-q^{-c(n)}[c(n)]_{q}) E_{ii}^{\lambda} \nonumber \\ &\qquad\qquad\qquad=q^{-2(c(1)+\cdots+c(n))}M_{\bar{1}} \cdots M_{\bar{n}}D_{\bar{n}}\cdots D_{\bar{1}} E_{ii}^{\lambda}. \end{aligned}
\end{equation}
\tag{5}
$$
In [1] the quantum matrix Capelli identities (5) were established for the special cases of one-column and one-row Young diagrams. This result was subsequently generalized to arbitrary Young diagrams in [4] for reflection equation algebras associated with Drinfeld–Jimbo $R$-matrices. In this framework quantum immanants were introduced. They are q-analogues of the immanants introduced by Okounkov. The universal matrix Capelli identity (1) holds universally for all reflection equation algebras, regardless of the form of the $R$-matrix and Young diagrams. If the $R$-matrix admits an $R$-trace, then quantum immanants can explicitly be constructed by applying the $R$-traces $\operatorname{Tr}_{R(12\ldots n)}$ in all tensor components to the left-hand side of (5).
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\Bibitem{Zai25} |
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