Example: Symplectic Monoids

Let $n=2m$ be even. We will apply the FRT-construction to two endomorphisms $\beta, \gamma \in {\cal E}_{2}={\cal E}\otimes {\cal E}$. In order to define them we have to introduce some notation. First consider the involution $i':=n-i+1$ on $\underline{n}$, that is

$$(1',2', \ldots , n')=(n,n-1, \ldots , 1).$$

Further set $\epsilon_i:=1$ if $i\leq m$ and $\epsilon_i:=-1$ if $i>m$ and define

$$\beta :=\sum_{i,j\in \underline{n}}e_{i}^{ j} \otimes e_{j}^{ i}$$

$$\gamma :=\sum_{i,j\in \underline{n}} \epsilon_i \epsilon_j e_{i}^{ j'} \otimes e_{i'}^{ j}.$$

The first endomorphism is just the flip operator on $V \otimes V$, whereas the second is an integer multiple of a projection map whose kernel is just the kernel of the linear form on $V \otimes V$ corresponding to the canonical skew bilinear form on $V$ (see below) and whose image is the one dimensional span of a skew bivector. Our object of interest will be the FRT-construction

$$A^{{\rm s}}_{R}(n) :={\cal M}(\{\beta, \gamma\}).$$

According to the preceeding section it is a graded matric bialgebra whose homogenous summands

$$A^{{\rm s}}_{R}(n,r):=M({\cal A}_r)$$

are the centralizer coalgebras of algebras ${\cal A}_r$ which are generated by endomorphisms

$$\beta_i := {\rm id}_{V^{\otimes i-1}} \otimes \beta \otimes{\rm id}_{V^{\otimes r-i-1}},\in {\cal E}_r$$

$$\gamma_i :={\rm id}_{V^{\otimes i-1}} \otimes \gamma \otimes{\rm id}_{V^{\otimes r-i-1}} \in {\cal E}_r$$

for $i=1, \ldots , r-1$. Using the notation of [We] the Brauer centralizer algebra $D_{r}({x})$ ($x$ an element in $R$) is generated by symbols $g_i$ and $e_i$ for $i=1, \ldots , r-1$ and the assignment $g_i\mapsto \beta_i$ and $e_i\mapsto \gamma_i$ defines a representation of $D_{r}({-n})$ on $V^{\otimes r}$. Thus ${\cal A}_r$ is just the image of $D_{r}({-n})$ under this representation, in H. Wenzl's notation from [We]: ${\cal A}_r=B_r({\rm Sp}_{}(n))$. R. Brauer showed in [Br] that this is just the centralizer algebra of the symplectic group ${\rm Sp}_{{\mathbb{C}}}(n)$ acting on $V^{\otimes r}$ if $R={\mathbb{C}}$. One of our aims is to generalize this to the case of an arbitrary algebraically closed field $K$ instead of ${\mathbb{C}}$.

Another problem, connected with the former, is to show that the centralizer algebra $C({\cal A}_r)$ of ${\cal A}_r$ which will turn out to be the symplectic Schur algebra $S_0(n,r)$ defined by S. Donkin in [Do2] is stable under base changes. In view of theorem 4.3 this is equivalent to the projectivity of the coalgebra $A^{{\rm s}}_{R}(n,r)$ as an $R$-module. For this purpose we are going to construct a basis for the latter one. The procedure follows [Oe] where the more general quantum case is treated. But it will become more transparent in the much simpler classical case. First note that there is an epimorphism of graded bialgebras from

$$A_{R}(n):=R[x_{1 1}, x_{1 2}, \ldots , x_{n n}]$$

to $A^{{\rm s}}_{R}(n)$ leaving the symbols $x_{i j}$ fixed (we use $x_{i j}$ as symbols for residue classes of ${{e}^*_{i}}^{j}$ in all cases of matric bialgebras). For $A_{R}(n)$ is just the FRT-construction ${\cal M}(\beta)$ where the relations coming from the endomorphism $\gamma$ are omitted. This is because $x_{i_2 j_1}x_{i_1 j_2}=\beta x_{{\bf i} {\bf j}}=x_{{\bf i} {\bf j}}\beta=x_{i_1 j_2}x_{i_2 j_1}$ just give the ordinary commutativity relations. The kernel of this bialgebra epimorphism is the ideal in $A_{R}(n)$ which is generated by the polynomials $\gamma x_{{\bf i} {\bf j}}=x_{{\bf i} {\bf j}}\gamma$ where ${\bf i},{\bf j} \in I(n,2)$. To write down these polynomials explicitly let us fix some notation:

$$f_{ij}:=\sum_{k=1}^n \epsilon_kx_{i k}x_{j k'},\;\; \bar f_{ij}:=\sum_{k=1}^n \epsilon_kx_{k i}x_{k' j} \in A_{R}(n,2).$$

Setting ${\bf i}=(i,j)$ and ${\bf j}=(k,l)$ we obtain

$$\gamma x_{{\bf i} {\bf j}} = \left\{\begin{array}{ll} \epsilo... ...psilon_l f_{ij} & k=l' \\ 0 & k \neq l' \end{array} \right .$$

Therefore we have

$$A^{{\rm s}}_{R}(n)=A_{R}(n)/{\cal F}$$ (13)

where ${\cal F}$ is the ideal in $A_{R}(n)$ generated by the set

$$F:=\{ f_{ij},\bar f_{ij}, f_{ll'}-\bar f_{kk'}\vert\; 1 \leq i <j\leq n, i\neq j', \; 1 \leq l\leq k \leq m\}.$$ (14)

If $R=K$ is an algebraically closed field we can interpret this in terms of algebraic geometry, i.e. we can look at the vanishing set of ${\cal F}$ in the monoid ${\rm M}_{K}(n)$ of $n\times n$-matrices. It is easy to see that this is just the closed submonoid

$${\rm SpM}_{K}(n):= \{ A \in {\rm M}_{K}(n)\vert\;\exists \;\; d(A)\in K,\; A^tJA=AJA^t=d(A)J\}$$

in ${\rm M}_{K}(n)$ called the symplectic monoid by S. Doty [Dt] and which has been considered by D.J. Grigor'ev [Gg] first. Here $J$ is the Gram-matrix of the canonical skew bilinear form, that is $J=(J_{ij})_{i,j \in \underline{n}}$ where $J_{ij}:=\epsilon_i\delta_{ij'}$. The function $d:{\rm SpM}_{K}(n) \rightarrow K$ is called the coefficient of dilation. It is neccesarily a regular function on ${\rm SpM}_{K}(n)$ and already well defined in $A^{{\rm s}}_{K}(n)$, explicitly:

$$d= \epsilon_k f_{kk'}= \epsilon_k\bar f_{kk'} \in A^{{\rm s}}_{R}(n,2).$$ (15)

Note that this is independent of $k\in \underline{n}$ by the relations in $A^{{\rm s}}_{K}(n)$. Furthermore $d$ is a group-like element of this bialgebra (cf. [Oe] 2.1.1). The set ${\rm GL}_{K}(n) \cap {\rm SpM}_{K}(n)$ of invertible elements in ${\rm SpM}_{K}(n)$ is precisely the group ${\rm GSp}_{K}(n)$ of symplectic similitudes. S. Doty showed in [Dt] that ${\rm SpM}_{K}(n)$ in fact coincides with the Zariski-closure of ${\rm GSp}_{K}(n)$ in ${\rm M}_{K}(n)$. We will obtain this as an easy consequence of the results presented below.

To write down a basis for $A^{{\rm s}}_{R}(n,r)$ we need some combinatorics. The set of partitions of $r$ is denoted by $\Lambda^+(r)$. It contains subsets $\Lambda^+(l, r)$ which consist of partitions having not more than $l$ parts. We write partitions as $l$-tuples $\lambda :=(\lambda_1, \lambda_2, \ldots , \lambda_l)$ of nonnegative integers $\lambda_i$ in descending order $\lambda_1\geq \lambda_2 \geq \ldots \geq \lambda_l\geq 0$ such that $\lambda_1 +\ldots +\lambda_l=r$. To each partition one associates a Young-diagram reading row lengths out of the components $\lambda_i$. For example

$$\begin{array}{*{4}{\vert p{2mm}}\vert} \cline{1-3} & & &\mu... ...\ \cline{1-2} &\multicolumn{3}{c}{} \ \cline{1-1}\end{array}$$

is associated to $\lambda=(3,2,2,1) \in \Lambda^+(4, 8)$. The column lengths of the diagram lead to another partition $\lambda'\in \Lambda^+(\lambda_1, r)$ called the dual of the partition $\lambda$, i.e. $\lambda'_i:=\vert\{j\vert\; \lambda_j \geq i\}\vert$. Let ${\cal S}_{r}$ denote the symmetric group on $r$ symbols and ${\cal S}_{\lambda}$ the standard Young subgroup of ${\cal S}_{r}$ corresponding to the partition $\lambda$. This is the subgroup fixing the subsets $\{1, \ldots , \lambda_1 \}, \; \{\lambda_1+1, \ldots , \lambda_1+\lambda_2\}, \ldots$ of $\underline{r}$. In the above example $\lambda=(3,2,2,1)$ the standard Young subgroup of ${\cal S}_{8}$ corresponding to the dual partition $\lambda'$ fixes $\{1,2,3,4\},\;\{5,6,7\}$ and $\{8\}$.

To each partition $\lambda\in \Lambda^+(r)$ and a pair of multi-indices ${\bf i},{\bf j}\in I(n,r)$ one defines a bideterminant $T^{\lambda}({\bf i}:{\bf j})\in A_{R}(n,r)$ by

$$T^{\lambda}({\bf i}:{\bf j}):= \sum_{w \in {\cal S}_{\lambda... ... \in {\cal S}_{\lambda'}} {\rm sign}(w)x_{({\bf i}w){\bf j} .}$$

where ${\bf i}w:=(i_{w(1)}, i_{w(2)}, \ldots , i_{w(r)})$. These are products of minor determinants, one factor for each column, the size of which correspond to the length of the column. By (13) they can be interpreted as elements of $A^{{\rm s}}_{R}(n,r)$, as well. We wish to write down a basis of the latter $R$-module consisting of such bideterminants. Since they are too large in number one needs a criterion to single out the right ones. This can be done using $\lambda$-tableaux. These are constructed from the diagram of $\lambda$ by inserting the components of a multi-index column by column into the boxes. In the above example:

$$T^{\lambda}_{{\bf i}}:= \begin{array}{*{4}{\vert p{2mm}}\ver... ...e{1-2} $i_4$ &\multicolumn{3}{c}{} \ \cline{1-1}\end{array}.$$

We put a new order $\ll$ on the set $\underline{n}$, namely $1'\ll 1 \ll 2' \ll 2 \ll \ldots \ll m' \ll m$. A multi-index ${\bf i}$ is called $\lambda$-column standard if the entries in $T^{\lambda}_{{\bf i}}$ are strictly increasing down columns according to this order. It is called $\lambda$-row standard if the entries in $T^{\lambda}_{{\bf i}}$ are weakly increasing along rows and $\lambda$-standard if it is both at the same time. We write $I_{\lambda}^{}$ to denote the subset of $I(n,r)$ consisting of all multi-indices being $\lambda$-standard. Such a multi-index ${\bf i}\in I_{\lambda}^{}$ is called $\lambda$-symplectic standard if for each index $i\in \underline{m}$ the occurences of $i$ as well as $i'$ in $T^{\lambda}_{{\bf i}}$ is limited to the first $i$ rows. The corresponding subset of $I_{\lambda}^{}$ will be denoted by $I_{\lambda}^{\rm sym}$.

The notion of symplectic standard tableaux traces back to R.C. King [Ki] and it has appeard in a lot of work concerning symplectic groups and their representation theory (for details see [Do3]).

It is well known from invariant theory (cf. [Mr], section 2.5) that the collection of all bideterminants $T^{\lambda}({\bf i}:{\bf j})$ where $\lambda$ runs through $\Lambda^+(n, r)$ and ${\bf i}, {\bf j}$ run through $I_{\lambda}^{}$ form a basis of $A_{R}(n,r)$. Similarily we will prove in the next section

Theorem 6.1   The $R$-module $A^{{\rm s}}_{R}(n,r)$ has a basis given by

$${\bf B}_r:=\{ {d}^lT^{\lambda}({\bf i}:{\bf j})\vert\; 0\leq ... ...bda^+(m, r-2l),\; {\bf i},{\bf j} \in I_{\lambda}^{\rm sym}\}.$$

Before proving this let us have a look at some consequences. The first one generalizes theorem 9.5 (a) of [Dt] avoiding the restriction to characteristic zero. Furthermore, it contains corollary 5.5 (f) of that paper for algebraically closed fields.

Corollary 6.2   Let $K$ be an algebraically close field. Then $A^{{\rm s}}_{K}(n)$ coincides with the coordinate ring of the Zariski-closure $\overline{{\rm GSp}_{K}(n)}$ of ${\rm GSp}_{K}(n)$ in ${\rm M}_{K}(n)$. In particular ${\rm SpM}_{K}(n)$ is identical to $\overline{{\rm GSp}_{K}(n)}$. Therefore, a complete set of generators of the vanishing ideal of $\overline{{\rm GSp}_{K}(n)}$ in $A_{K}(n)$ is given by the set $F$ (defined in equation 14).

PROOF: Let $A_0(n)$ be the coordinate ring of $\overline{{\rm GSp}_{K}(n)}$ and $A_0(n,r)$ its $r$-th homogenous summand. In [Do2] the symplectic Schur algebra $S_0(n,r)$ is defined as the dual algebra to the coalgebra $A_0(n,r)$. The dimension of the latter one is given by Weyl's character formula and therefore independent of the field $K$ (cf. [Do2] p. 77). On the other hand there is an epimorphism of graded bialgebras from $A^{{\rm s}}_{K}(n)$ to $A_0(n)$ since $\overline{{\rm GSp}_{K}(n)}$ is closed in ${\rm SpM}_{K}(n)$ and the latter one has been defined as the vanishing set of the ideal ${\cal F}$ by which $A^{{\rm s}}_{K}(n)$ is defined. But by our basis theorem 6.1 the dimension of $A^{{\rm s}}_{K}(n,r)$ is independent of the field $K$, as well. Thus, the proof can be finished looking at the case $K={\mathbb{C}}$ and using Doty's theorem 9.5 (a) or alternately by a direct calculation of $\vert{\bf B}_r\vert={\rm dim}_{{\mathbb{C}}}{((A_0(n,r)))}$ (see proposition 7.1 below).$\Box$

By theorem 4.1 we have isomorphisms

$$K\otimes_{{\mathbb{Z}}}A^{{\rm s}}_{{\mathbb{Z}}}(n,r) \cong ... ...hbb{Z}}}A^{{\rm s}}_{{\mathbb{Z}}}(n) \cong A^{{\rm s}}_{K}(n).$$

Since $A^{{\rm s}}_{K}(n)$ has been recognized to be the coordinate ring of ${\rm SpM}_{K}(n)=\overline{{\rm GSp}_{K}(n)}$ we may interpret the spectrum of the ring $A^{{\rm s}}_{{\mathbb{Z}}}(n)$ as an integral monoid scheme ${\rm SpM}_{{\mathbb{Z}}}(n)$. Accordingly, an integral form for the symplectic Schur algebra can be obtained as the dual algebra

$$S_{{\mathbb{Z}}}^{\rm s}(n,r):={\rm Hom}_{{\mathbb{Z}}}{(A^{{\rm s}}_{{\mathbb{Z}}}(n,r),{\mathbb{Z}})}$$

of its homogenous summands. By theorem 4.3 and 6.1 this is stable under base change, that is, tensoring by a field $K$ gives the symplectic Schur algebra $S_{K}^{\rm s}(n,r)=S_0(n,r)={\rm Hom}_{K}{(A^{{\rm s}}_{K}(n,r),K)}$ defined over that field. An integral form for symplectic Schur algebras exists, as well, by S. Donkin's work on generalized Schur algebras (see [Do2]). But his approach is quite different using the theory of Lie algebras in particular the Kostant ${\mathbb{Z}}$-form. In both cases the notion of symplectic Schur algebras can be extended to more general integral domains $R$ instead of ${\mathbb{Z}}$ leading to identical concepts. In our case this is $S^{\rm s}_{R}(n,r):={\rm Hom}_{R}{(A^{{\rm s}}_{R}(n,r),R)}$. By theorem 3.3 we conclude

Corollary 6.3   Over any noetherian integral domain $R$ the symplectic Schur algebra is isomorphic to the centralizer algebra of the Brauer algebra $D_{r}({-n})$.

For a field of characteristic zero this has been proved by S. Doty, too ([Dt] corollary 9.3. (c)). It should be remarked, that the basis dual to ${\bf B}_r$ together with the anti-involution defined by matrix transposition give a cell datum for the symplectic Schur algebra in the sense of J. Graham and G. Lehrer (cf. [Oe], 4.2.5). Thus, its representation theory can be developed easily to the extent of the treatment of cellular algebras in [GL].