Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ variables. Then $\textrm{GL}_n$ acts on $K[x_{ijk}]$ by simultaneous conjugation $$ (X_1,\ldots,X_d) \mapsto (gX_1g^{-1},\ldots,gX_dg^{-1}) , \ \ \ \ \ \ \ \ g \in \textrm{GL}_n $$ I am just beginning to try to understand the theory of matrix invariants $K[x_{ijk}]^{\textrm{GL}_n}$, for instance with the Drenksy's survey COMPUTING WITH MATRIX INVARIANTS and Procesi's The invariant theory of n × n matrices. In the latter we have

• The algebra $K[x_{ijk}]^{\textrm{GL}_n}$ is generated by $\textrm{tr}(X_{k_1}\cdots X_{k_m})$ for products of $X_1,\ldots,X_d$
• The algebra $K[x_{ijk}]^{O(n)}$ for $O(n) \subseteq \textrm{GL}_n$ is generated by the trace $\textrm{tr}$ evaluated on products involving $X_k$ and $X_k^t$.
• The algebra $K[x_{ijk}]^{U(n)}$ for $U(n) \subseteq \textrm{GL}_n$ is generated by the trace $\textrm{tr}$ evaluated on products involving $X_k$ and $X_k^*$.

and many more results involving minimal generators and relations.

My question is, what is the analogous scenario for $S_n \subseteq \textrm{GL}_n$ where $S_n$ is the finite subgroup of permutation matrices? Perhaps the analysis is much easier than the other cases, but I do not have a good reference.

An explicit description of these invariants can be found in Male's work on permutation-invariant random matrices using an operad, but there is no reference to this problem there. There seems to be some connection with random matrices invariant by a subgroup of $\textrm{GL}_n$, and the evaluation (in expectation and in the limit) of its invariants. In particular, it would be interesting to know when these invariants are described by an operad, or if this is unique to $S_n$.

Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ variables. Then $\textrm{GL}_n$ acts on $K[x_{ijk}]$ by simultaneous conjugation $$ (X_1,\ldots,X_d) \mapsto (gX_1g^{-1},\ldots,gX_dg^{-1}) , \ \ \ \ \ \ \ \ g \in \textrm{GL}_n $$ I am just beginning to try to understand the theory of matrix invariants $K[x_{ijk}]^{\textrm{GL}_n}$, for instance with the Drenksy's survey COMPUTING WITH MATRIX INVARIANTS and Procesi's The invariant theory of n × n matrices. In the latter we have

• The algebra $K[x_{ijk}]^{\textrm{GL}_n}$ is generated by $\textrm{tr}(X_{k_1}\cdots X_{k_m})$ for products of $X_1,\ldots,X_d$
• The algebra $K[x_{ijk}]^{O(n)}$ for $O(n) \subseteq \textrm{GL}_n$ is generated by the trace $\textrm{tr}$ evaluated on products involving $X_k$ and $X_k^t$.
• The algebra $K[x_{ijk}]^{U(n)}$ for $U(n) \subseteq \textrm{GL}_n$ is generated by the trace $\textrm{tr}$ evaluated on products involving $X_k$ and $X_k^*$.

and many more results involving minimal generators and relations.

My question is, what is the analogous scenario for $S_n \subseteq \textrm{GL}_n$ where $S_n$ is the finite subgroup of permutation matrices? Perhaps the analysis is much easier than the other cases, but I do not have a good reference.

An explicit description of these invariants can be found in Male's work on permutation-invariant random matrices, but there is no reference to this problem there. The invariants can be generated by what are called there the injective generalized trace $\tau^0$ defined on edge-labeled graphs.

What's interesting is that the equivariant maps on permutation-invariant (random) matrices form an operad, so there is a nice view of such maps and invariants. A broader question is whether equivariant maps/invariants for other finite subgroups of $\textrm{GL}_n$ also have some kind of operadic description, or if this is unique to $S_n$.

Let $K$ be a field of characteristic 0, and consider $d$ generic matrices $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ variables. Then $\textrm{GL}_n$ acts on $K[x_{ijk}]$ by simultaneous conjugation $$ (X_1,\ldots,X_d) \mapsto (gX_1g^{-1},\ldots,gX_dg^{-1}) , \ \ \ \ \ \ \ \ g \in \textrm{GL}_n $$ I am just beginning to try to understand the theory of matrix invariants, that is aspects of $K[x_{ijk}]^{\textrm{GL}_n}$, for instance with the Drenksy's survey COMPUTING WITH MATRIX INVARIANTS and Procesi's The invariant theory of n × n matrices. In the latter we have

• The algebra $K[x_{ijk}]^{\textrm{GL}_n}$ is generated by $\textrm{tr}(X_{k_1}\cdots X_{k_m})$ for products of $X_1,\ldots,X_d$
• The algebra $K[x_{ijk}]^{O(n)}$ for $O(n) \subseteq \textrm{GL}_n$ is generated by the trace $\textrm{tr}$ evaluated on products involving $X_k$ and $X_k^t$.
• The algebra $K[x_{ijk}]^{U(n)}$ for $U(n) \subseteq \textrm{GL}_n$ is generated by the trace $\textrm{tr}$ evaluated on products involving $X_k$ and $X_k^*$.

and many more results involving minimal generators and relations.

My question is, what is the analogous scenario for $S_n \subseteq \textrm{GL}_n$ where $S_n$ is the finite subgroup of permutation matrices? Perhaps the analysis is much easier than the other cases, but I do not have a good reference.

An explicit description of these invariants can be found in Male's work on permutation-invariant random matrices using an operad, but there is no reference to this problem there. There seems to be some connection with random matrices invariant by a subgroup of $\textrm{GL}_n$, and the evaluation (in expectation and in the limit) of its invariants. In particular, it would be interesting to know when these invariants are described by an operad, or if this is unique to $S_n$.

Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ variables. Then $\textrm{GL}_n$ acts on $K[x_{ijk}]$ by simultaneous conjugation $$ (X_1,\ldots,X_d) \mapsto (gX_1g^{-1},\ldots,gX_dg^{-1}) , \ \ \ \ \ \ \ \ g \in \textrm{GL}_n $$ I am just beginning to try to understand the theory of matrix invariants $K[x_{ijk}]^{\textrm{GL}_n}$, for instance with the Drenksy's survey COMPUTING WITH MATRIX INVARIANTS and Procesi's The invariant theory of n × n matrices. In the latter we have

• The algebra $K[x_{ijk}]^{\textrm{GL}_n}$ is generated by $\textrm{tr}(X_{k_1}\cdots X_{k_m})$ for products of $X_1,\ldots,X_d$
• The algebra $K[x_{ijk}]^{O(n)}$ for $O(n) \subseteq \textrm{GL}_n$ is generated by the trace $\textrm{tr}$ evaluated on products involving $X_k$ and $X_k^t$.
• The algebra $K[x_{ijk}]^{U(n)}$ for $U(n) \subseteq \textrm{GL}_n$ is generated by the trace $\textrm{tr}$ evaluated on products involving $X_k$ and $X_k^*$.

and many more results involving minimal generators and relations.

My question is, what is the analogous scenario for $S_n \subseteq \textrm{GL}_n$ where $S_n$ is the finite subgroup of permutation matrices? Perhaps the analysis is much easier than the other cases, but I do not have a good reference.

An explicit description of these invariants can be found in Male's work on permutation-invariant random matrices using an operad, but there is no reference to this problem there. There seems to be some connection with random matrices invariant by a subgroup of $\textrm{GL}_n$, and the evaluation (in expectation and in the limit) of its invariants. In particular, it would be interesting to know when these invariants are described by an operad, or if this is unique to $S_n$.