数理解析研究所講究録 1262 巻 2002 年 30-51

30
SPIN REPRESENTATIONS AND CENTRALIZERALGEBRAS FOR Spin(2n)KAZUHIKO KOIKEAOYAMAGAKUIN UNIVERSITY

DEPT. OF MATH.

青山学院大学理工学部 小池和彦
1. INTRODUCTION

In this exposition, we deal with the even spin case.
Letbe Spin(2n, ) or Pin(2n, ), namely the double coveringgroup of SO(2n) or $O(2n)$ .
Then every irreducible representation of , not coming from the reresentations of SO or can be realized in the tensor spacefor some , whereis the fundamental spin representation of andis the natural representation of $O(n)$ .
In this case we mainly deal with the Pin(2n) centralizer algebra$G$$\mathbb{C}$$\mathbb{C}$

$G$

-

$O$

$k$

$\Delta^{(\pm)}\otimes\otimes^{k}V$$\Delta^{(\pm)}$$G$$V=\mathbb{C}^{n}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}=\mathrm{H}\mathrm{o}\mathrm{m}_{P\cdot n(2n)}.$(,$\Delta\otimes\otimes^{k}V$a$\otimes\otimes^{k}V$)and define two kinds of basis of this algebra for $k\leqq n$ , just as in
the case of Spin(2n+1). The argument goes well for this algebra
similarly as that of Spin(2n +1). Finally we consider the associaterof the irreducible representations
and,where $‘\det$ ’ denotes the linear representation of Pin(2n) induced fromthe natural homomorphism Pin(2n) $arrow O(2n)arrow\det\{\pm 1\}$ .
This associater is given by the endomorphism
whereis the associater forand is given by the degree opera. This associater commutes with the action of Spin(2n) and if it is restricted to the irreducible representationof Pin(2n), it becomes an associater ofand its-eigenspaces give the irreducible representationof Spin(2n) respectively. So we can pickup the irreducible representation of Spin(2n).
$\Delta\otimes\otimes^{k}V$$\Delta\otimes\otimes^{k}V\otimes \mathrm{d}\mathrm{e}\mathrm{t}$$(A\otimes \mathrm{i}\mathrm{d})\in \mathrm{E}\mathrm{n}\mathrm{d}(\Delta\otimes\otimes^{k}V)$$A$$\Delta$$\mathrm{t}\mathrm{o}\mathrm{r}A[i_{1},i_{2}, i_{3}, \ldots, i_{r}]=(-1)^{r}[i_{1}, i_{2}, i_{3}, \ldots, i_{r}]$$[\Delta, \delta]$$[\Delta,\cdot\delta]\cong[\Delta, \delta]\otimes\det$$\{\pm 1\}$$\mathrm{s}(1/2+\delta)_{S\dot{\mu}n(2n)}^{\pm}$

2. A SUMMARY OF REPRESENTATION OF Spin(2n) AND Pin(2n)We first fix notations. As the defining symmetric matrix of $O(2n)$ ,we take an anti-diagonal matrix $S=(\delta:,2n+1-:)$ of size $2n$ .
The setof the dominant integral weights for Lie algebra so
is given by$P^{+}$$(2n, \mathbb{R})$$P^{+}=\{\lambda_{1}\epsilon_{1}+\lambda_{2}\epsilon_{2}+\ldots+\lambda_{n}\epsilon_{n};\lambda_{1}\geq\lambda_{2}\geq\ldots\geq|\lambda_{n}|\geq 0\}$,

31
Kazuhiko Koike$,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$all the$\lambda_{i}’ \mathrm{s}$are integers or half-integers (namely$1/2+\mathbb{Z}$) simplytaneously.
denote the irreducible representation (or character) withLet.
A
If all the Aj’s are integers, the corresponding irreducible representation comes from that of SO(2n) and we also write XsO{2n) instead of$\lambda_{Spin(2n)}$$\in P^{+}$$\lambda_{Spin(2n)}$.

is the vector representation (the natural representaThen (1)ofis the irreducible representationtion) and
1, . . . ’ $n-1$ ).
is not irreducible and $\wedge^{n}V=(1^{n})so(2n)+(1^{n-1}, -1)\mathrm{S}0(2\mathrm{n})$For SO $(2n),$, $n-1$
by for $i=1,2$ ,As usual, we denote this representation
and $e_{n}^{+}=(1^{n})_{SO(2n)}$ and $e_{n}^{-}=(1^{n-1}, 1)_{SO(2n\rangle}$ .
. For apartition
We note that
we denote the irreducible representations by$\wedge^{i}V$$(1^{i})_{SO(2n)}$$\epsilon \mathit{0}(2n)(i=$$\wedge^{n}V$$\wedge^{i}V$$e_{i}$$\wedge^{i}V\cong\wedge^{2n-i}$ $V$$\ldots$$\delta=(\delta_{1}, \delta_{2}, \ldots, \delta_{n})$$(1/2+\delta)_{Spin(2n)}^{+}=(1/2+\delta_{1},1/2+\delta_{2}, \ldots, 1/2+\delta_{n})_{Spin(2n)}$and$(1/2+\delta)_{Spin(2n)}=(1/2+\delta_{1},1/2+\delta_{2}, \ldots, 1/2+\delta_{n-1}, -1/2-\delta_{n})_{Spin(2n)}$.

We put simply $\Delta^{+}=(1/2+\emptyset)_{Spin(2n)}^{+}$ and $\Delta^{-}=(1/2+\emptyset)_{Spin(2n)}$ .
and
We also put$(1/2+\delta_{1},1/2+\delta_{2},\ldots, 1/2+\delta_{n})\in P^{+}$, we introduce theFor A
sum character and the difference character for Spin(2n) and denotethem by$\Delta=\Delta^{+}+\Delta^{-}$$\Delta’=\Delta^{+}-\Delta^{-}$$=$$[\Delta, \delta]_{Spin(2n)}=\lambda_{Spin(2n)}^{(+)}=(1/2+\delta)_{Spin(2n)}^{+}+(1/2+\delta)_{Spin(2n)}$and$[\Delta’, \delta]_{Spin(2n)}=\lambda_{Spin(2n)}^{(-)}=(1/2+\delta)_{Spin(2n)}^{+}-(1/2+\delta)_{Spin(2n)}$.

becomes the irreducible character of Pin(2n).
Then, we denote the sum
Similarly for apartition $\mu=$bySO(2n)
forcharacterthedifference
character and$[\Delta, \delta]_{Spin(2n)}$$(\mu_{1}, \mu_{2}, \ldots, \mu_{n})$$\mu_{SO(2n)}^{(+)}=(\mu_{1}, \mu_{2}, \ldots, \mu_{n})_{SO(2n)}+(\mu_{1}, \ldots, \mu_{n-1}, -\mu_{n})_{SO(2n)}$and$\mu_{SO(2n)}^{(-)}=(\mu_{1}, \mu_{2}, \ldots, \mu_{n})_{SO(2n)}-(\mu_{1}, \ldots, \mu_{n-1}, -\mu_{n})_{SO(2n)}$.

with $\ell(\mu)<n$ , $\mu SO(2n)$ becomes an irreducible
For apartitionin thisforcharacter of $O(2n)$ , so we sometimes write$\mu$$\mu_{O(2n)}$case.

If $\ell(\mu)=n$ ,sometimes write$\mu_{SO(2n)}^{(+)}$$\mu O(2n)$$\mu_{SO(2n)}$is the irreducible character of $O(2n)$ and we.
for$\mu_{SO(2n)}^{(+)}$

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Centralizer algebras for even SpinFor$\epsilon\in\{\pm 1\}$

, if $\epsilon=+1$ , we put, etc. and if $\epsilon=-1$ ,, etc.

$\Delta^{\epsilon}=\Delta^{+}$$(1/2+\delta)_{S\dot{\mu}n(2n)}^{+}$,$e_{n}^{\epsilon}=e_{n}^{+}$$\Delta^{\epsilon}=\Delta^{-}$,,$(1/2+\delta)_{S\dot{\mu}n(2n)}^{\epsilon}=$$e_{n}^{\epsilon}=e_{n}^{-}$, $(1/2+$$\delta)_{S\dot{\mu}n(2n)}^{\epsilon}=(1/2+\delta)_{S\dot{\mu}n(2n)}$

Theorem 2.1. For , ,$\epsilon$$\epsilon_{1}$$\epsilon_{2}\in\{\pm 1\}$(i)

(2.1.1), we have the following formulas.

$( \Delta^{\epsilon})^{2}=e_{n}^{\epsilon}+\sum_{\dot{|}=0}^{[n/21}e_{n-2:}$(2.1.2).

$\Delta^{+}\Delta^{-}=\sum_{\dot{l}=0}^{[(n-1)/2]}e_{n-1-2\dot{*}}$(ii) For a partition$\delta$.

with its length $\ell(\delta)<n$ , we have

(2.1.3)$(1/2+\delta)_{S\dot{\mu}n(2n)}^{\epsilon}(1)_{SO(2n)}$$=(1/2+\delta)_{S\dot{\mu}n(2n)}^{\epsilon}+$$| \mu/\delta|=’ 1\sum_{\mu\supset\delta}(1/2+\mu)_{S\dot{\mu}n(2n)}^{\epsilon}+$ $| \delta/\mu|=’ 1\sum_{\delta\supset\mu}(1/2+\mu)_{S\dot{\mu}n(2n)}^{\epsilon}$If$\ell(\delta)=n$.

, we have$(1/2+\delta)_{S\dot{\mu}n(2n)}^{\epsilon}(1)_{SO(2n)}$(2.1.4).

$=$$| \delta/\mu|=’ 1\sum_{\delta\supset\mu}(1/2+\mu)_{S\dot{\mu}n(2n)}^{\epsilon}$$| \mu/\delta|-,1\ell(\mu)\overline{\leqq}n\sum_{\mu\supset\delta},(1/2+\mu)_{S\dot{\mu}n(2n)}^{\epsilon}+$(iii) For a partition$\mu$with $\ell(\mu)<n$ , we have$(2.1.5)$$\Delta^{\epsilon}\mu_{SO(2n)}=$$(1/2+\nu)_{S\dot{\mu}n(2n)}^{\epsilon}+$$(1/2+\nu)_{S\dot{\mu}n(2n)}^{\epsilon}$$\sum_{\mu\supseteq\nu}$$\sum_{\mu\supseteq\nu}$$\mu/\nu$vertical strip$|\mu/\nu|\equiv 0$$\mu/\nu$$\mathrm{m}\mathrm{o}\mathrm{d} 2$vehioel str.p$|\mu/\nu|\equiv 1$$\mathrm{m}\mathrm{o}\mathrm{d} 2$(iv) For a part ition $\mu=(\mu_{1}, \mu_{2}, \ldots,\mu_{n})$ with $\ell(\mu)=n$ , we write$\mu_{SO(2n)}^{+}=\mu_{SO(2n)}$ and. Thenwe have$\mu_{S\mathrm{O}(2n)}^{-}=(\mu_{1}, \ldots, \mu_{n-1}, -\mu_{n})_{SO(2n)}$(2.1.6).
$\Delta^{\epsilon_{1}}\mu_{SO(2n)}^{\epsilon_{2}}=$$(1/2+\nu)_{S\dot{\mu}n(2n)}^{\epsilon_{2}}$$\sum_{\mu\supseteq\nu}$$\mu/\nu veni\omega lst\dot{n}p\mathrm{t}-1)^{|\mu/|=\epsilon_{1}\epsilon_{2}}$

.

.

33
Kazuhiko Koike

Particularly, for the exterior products$e_{i}$’s if $i<n$ , we have$\Delta^{\epsilon}(1^{i})_{SO(2n)}=\sum_{0\leqq 2s\leqq i}\langle 1/2+(1^{i-2s}))_{Spin(2n)}^{\epsilon}+\sum_{0\leqq 2s\leqq i-1}(1/2+(1^{i-1-2s}))_{Spin(2n)}^{\epsilon}$

and for$e_{n}^{\epsilon}$, we have$\Delta^{\epsilon}e_{n}^{\epsilon}=\sum_{0\leqq 2s\leqq n}(1/2+(1^{n-2s}))_{Spin(2n)}^{\epsilon}$and.
$\Delta^{-\epsilon}e_{n}^{\epsilon}=\sum_{0\leqq 2s\leqq n-1}(1/2+(1^{n-1-2s}))_{Spin(2n)}^{\epsilon}$(v),$(e_{n}^{+})^{2}=s$$(e_{n}^{+})(e_{n}^{-})= \sum_{s+t\leqq n-1}(2^{s}, 1^{t})_{SO(2n)}s--n-1\mathrm{m}\mathrm{o}\mathrm{d} 2\overline{t}\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} 2^{\cdot}$$t \overline{\equiv}0\mathrm{m}\mathrm{o}\mathrm{d} 2-n\mathrm{m}\mathrm{o}\mathrm{d} 20\leqq s+t\leqq n\sum_{-},(2^{s}, 1^{t})_{SO(2n)}$$\sum$$(2^{s}, 1^{t})_{SO(2n)}$$s \leqq n-2\sum_{\mathrm{m}\mathrm{o}\mathrm{d} 2},(2^{s}, 1^{n-s-1}, -1)_{SO(2n)}+$$(e_{n}^{-})^{2}=(2^{n-1}, -2)_{SO(2n)}+s\equiv n$$s–n\mathrm{m}\mathrm{o}\mathrm{d} 2t\overline{\equiv}0\mathrm{m}\mathrm{o}\mathrm{d} 2s+t\leqq n-2$

From the above, we know all the irred spin representations occur in, so we have
. Also we havethe space$\Delta\otimes\otimes^{k}$$\oplus_{i=0}^{2n}\wedge^{l}V\cong\Delta\otimes(\Delta)^{*}$$V$

Lemma 2.2.
(2.2.1)$\mathrm{C}\mathrm{P}_{1}^{\mathrm{k}}=\mathrm{H}\mathrm{o}\mathrm{m}_{Pin(2n)}(\Delta\otimes\otimes^{k}V, \Delta\otimes\otimes^{l}V)\cong\oplus_{i=0}2^{ik+l}n((\wedge V)\otimes\otimes V)^{Pin(2n)}$Here the superscript Pin(2n) means the Pin(2n)-invariant space.
So we need to give an explicit Pin(2n)-equivariant isomorphism be-

.
and
tween
abaon
First we give an action of Lie(Pin(2n))
sis of , we take abasis parametrized by all the subsets of $[n]=$, where,and denote the basis elements by
1, 2,$(1\leqq i_{1}<i_{2}<\ldots<i_{r}\leqq n)$ just as before and we take aset of thesimple root vectors as follows.
$\oplus_{i=0}^{2n}\wedge^{i}V$$\Delta\otimes(\Delta)^{*}$$=\epsilon \mathrm{o}(2n, \mathbb{R})$$\Delta.\mathrm{A}\mathrm{s}$$\Delta$$\{$$\ldots$$\{[\mathrm{I}]\}$

$n\}$

,

$\mathrm{a}\mathrm{d}(X_{k})=E_{k,k+1}-E_{\overline{k+1},\overline{k}}$$\mathrm{a}\mathrm{d}(\mathrm{Y}_{k})=E_{k+1,k}-E_{\overline{k},\overline{k+1}}$$\mathrm{a}\mathrm{d}(h_{i})=E_{i,i}-E_{\overline{i},\overline{i}}$,$\mathrm{I}=\{i_{1}, i_{2}, \ldots, i_{r}\}$,$\mathrm{a}\mathrm{d}(X_{n})=E_{n-1,\overline{n}}-E_{n,\overline{n-1}}$,$\mathrm{a}\mathrm{d}(\mathrm{Y}_{n})=E_{\overline{n},n-1}-E_{\overline{n-1},n}$.

, where $k\in\{1,2, \ldots, n-1\}$ and $i\in\{1,2, \ldots, n\}$ .
on this base is given as
The explicit action of Lie algebra$\epsilon \mathrm{o}(2n, S)$.

34
Centralizer algebras for even Spin

Lemma 2.3.

ifand $k+1<$
otherwise,$-[i_{1}, \ldots, i_{s-1}, k+1, i_{s+1}, \ldots, i_{r}]$$X_{k}[i_{1}, i_{2}, \ldots, i_{r}]=\{^{\backslash }$$0$$k=i_{s}$and

if $i_{f-1}=n-1$ and $i_{f}=n$$-[i_{1}, i_{2}, \ldots, i_{f-2}]$$X_{n}[i_{1}, i_{2}, \ldots, i_{r}]=\{$

0otherwise,

if $k+1=i_{s}$ and $k>i_{s-1}$$-[i_{1}, \ldots, i_{s-1}, k, i_{s+1}, \ldots, i_{f}]$$\mathrm{Y}_{k}[i_{1}, i_{2}, \ldots, i_{r}]=\{$

0otherwise,and$\mathrm{Y}_{n}[i_{1}, i_{2}, \ldots, i_{r}]=\{\begin{array}{l}-[i_{1},i_{2},\ldots,i_{r},n-1,n]ifi_{r}<n-10othemise\end{array}$$h_{k}[i_{1}, i_{2}, \ldots, i_{r}]=\{.\frac{-1}{2}[_{1},i_{2},\ldots,i_{r}]\frac{\mathrm{l}}{2,i’}[i_{1},i_{2},..,i_{r}]$, where$othemiseifk\in\{i_{1},.i_{2}, \ldots, i_{r}\}$,, are in the increasing order.
Moreover let be a linear endomorphism of defined by the degree
operator. Then A becomesan associater of $(\Delta\otimes\det)\cong\Delta$ as Pin(2n) modules and on $\Delta=$,
is given by$i_{1}$,$i_{2}$$i_{r}$$\ldots$$A$$\Delta$$A[i_{1}, i_{2}, i_{3}, \ldots, i_{r}]=(-1)^{r}[i_{1}, i_{2}, i_{3}, \ldots, i_{r}]$$\Delta^{+}\oplus\Delta^{-}$$A$$A=\mathrm{i}\mathrm{d}_{\Delta+}\oplus-\mathrm{i}\mathrm{d}_{\Delta^{-}}$.

For any sequence , , ,, of non-negative integers, we define
the corresponding elementinsuch that it satisfys
the alternating property on the indices as before.
Acompact real formofisgenerated by the elements
and,, $(i=$
1, 2,, ) as a real Lie algebra.
Then the invariant hermitian forms of and under the action of
are given such thatbase $<u_{1}$ , ,, , ,,and the baseof become orthonormal basis respectively.
Then an explicit embedding theorem ofin the spaceisgiven as follows.
We write the-equivariant embedding fromtoby
. Here ,and we consider, as the notation, for example,
ifandand so on.
Theorem 2.4. If is an even positive integer, we haveandand only the embeddings
corresponding to the$i_{1}$$i_{2}$$i_{3}$$i_{r}$$\ldots$$[i_{1}, i_{2}, i_{3}, \ldots, i_{r}]$$\epsilon \mathrm{o}(2n)_{\varphi t}$$\sqrt{-1}h_{\dot{l}}$$\Delta$$L(Spin(2n))\otimes \mathbb{C}\cong \mathrm{s}\mathrm{o}(2n, \mathbb{C})$$\sqrt{-1}(X_{\dot{l}}+\mathrm{Y}_{\dot{l}})$$V$$\epsilon \mathrm{o}(2n)_{\varphi t}$$\Delta$$i\mathrm{h}\mathrm{e}$$u_{2}$$V$$X_{\dot{l}}-\mathrm{Y}_{\dot{l}}$$n$$\ldots$$[\mathrm{I}]_{\mathrm{I}\subset[n]}$$\ldots$$u_{n}$$u_{\overline{n}}$$\wedge^{:}V$$\epsilon \mathrm{o}(2n)$$\phi_{\ell}$$\phi_{\ell}^{\epsilon_{1}\epsilon_{2}}$$\epsilon_{1}$$u_{\overline{1}}>\mathrm{o}\mathrm{f}$$\Delta^{*}\otimes\Delta$$\wedge^{\ell}V$$\Delta^{\epsilon_{1}}\otimes(\Delta^{\epsilon_{2}})^{*}$$\epsilon_{2}\in\{\pm 1\}$$\phi_{\ell}^{\epsilon_{1}\epsilon_{2}}=\phi_{\ell}^{+-}$$\epsilon_{1}=1$$\epsilon_{2}=-1$$n$$(\Delta^{+})^{*}\cong\Delta^{+}$$(\Delta^{-})^{*}\cong\Delta^{-}$$\phi_{\ell}^{\epsilon_{1}\epsilon_{2}}$case$\ldots$$\Delta$$\epsilon_{1}=(-1)^{n-\ell}\epsilon_{2}$occurs35
Kazuhiko Koike

andis an odd positive integer, we have
case
the
to
corresponding
and only the embeddings

If$(\Delta^{+})^{*}\cong\Delta^{-}$$n$$(\Delta^{-})^{*}\cong\Delta^{+}$$\epsilon_{1}=-(-1)^{n-\ell}\epsilon_{2}$$\phi_{\ell}^{\epsilon_{1}\epsilon_{2}}$occurs.

$\frac{(-1)^{|\mathrm{W}-\mathrm{W}\cap \mathrm{K}|}}{2^{(n-|\mathrm{J}|-|\mathrm{I}|-1)/2}}[L4]$$\otimes[\mathrm{A}4]^{*}$$\phi_{\ell}^{\epsilon_{1}\epsilon_{2}}$$(<\mathrm{A}, \mathrm{A}, \overline{\mathrm{A}},\overline{\not\subset}>)=$$(-1)^{+k}.

\cdot=\epsilon_{1}\sum_{[n]-\mathrm{J}-\mathrm{I}\supseteq \mathrm{K}},$$(-l)^{j+k}=\epsilon_{2}$Here by small letters, we denote the number of the elements in theset indexed by their capital letters.
and , we have the followings.
For$e_{n}^{+}$$e_{n}^{-}$$\phi_{n}^{\epsilon_{1}\epsilon_{2}}(<\mathrm{A}, \mathrm{A},\overline{4-},\overline{arrow \mathrm{I}}>)(\epsilon_{1})=\sum_{=(-1)\mathrm{j}+k\epsilon_{1}(-1)=\epsilon_{2}}\frac{(-1)^{|\mathrm{w}-\mathrm{w}\mathrm{n}\kappa|}}{2^{(n-|\mathrm{J}|-|\mathrm{I}|-1)/2}}[_{-}\mathrm{b}^{\mathrm{K}}arrow]\otimes[\mathrm{A}^{X_{f}]^{*}}[n]-\mathrm{J}-\mathrm{I}\supseteq \mathrm{K}+k$’s becomes an isometric embedding
Moreover each of the above
with respect to the invariant hermitian inner products.
degree , we
, ,Here for an exterior productrespectively by
anddenotes the basis elements of$\phi_{\ell}^{\epsilon_{1}\epsilon_{2}}$$<\mathrm{A}$$e_{n}^{+}$$<\mathrm{A}$$\mathrm{A}$$n$$\overline{4_{-}},\overline{L}>\mathrm{o}\mathrm{f}$$e_{n}^{-}$,$\mathrm{A},\overline{<^{\mathrm{L}}},arrow^{\overline{\mathrm{I}}>}(+)=\frac{1}{\sqrt{2}}(<\mathrm{A}, \mathrm{A}, \overline{p_{-,\overline{\frac{r\mathrm{I}}{\backslash }}}}>+(-1)^{|\mathrm{I}|}<\mathrm{A}, \mathrm{b}^{\mathrm{c}},\overline{\mathrm{g}_{-}^{\mathrm{c}}},\overline{\mathrm{g}_{-}}>)$,$\mathrm{A},\overline{4_{-}},$

and$<\mathrm{A}$$\overline{\not\subset}>(-)=\frac{1}{\sqrt{2}}(<\mathrm{A}, \mathrm{A},\overline{4-},\succ^{\overline{\mathrm{I}}>}-(-1)^{|\mathrm{I}|}<\mathrm{A}, \mathrm{b}^{\mathrm{c}},\overline{\mathrm{A}^{\mathrm{c}}-},\overline{\mathrm{J}_{-}}>)$

the right-hand sides are non-zero.
Using the above, we obtain the following Pin(2n) equivariant embedding.
Theorem 2.5. As a Pin(2n) $(O(2n))$ module, for an even , theis given byequivariant isometric embedding$,\mathrm{i}\mathrm{f}$$\ell$$\phi_{\ell}$$:\wedge^{\ell}Varrow\Delta\otimes\Delta^{*}$$\phi_{\ell}(<\mathrm{A}, \mathrm{A},arrow’\overline{\not\subset}\overline{\mathrm{W}}>)=\sum_{\supset[n]-\mathrm{J}-\mathrm{I}_{arrow}\mathrm{K}}\frac{(-1)^{|\mathrm{w}-\mathrm{w}\mathrm{n}\kappa|}}{2^{(n-|\mathrm{J}|-|\mathrm{I}|)/2}}[\mathrm{A}4]$

and for an odd , the equivariant isometric embedding
is given by$\ell$$\phi\ell$$\otimes[\mathrm{A}4]^{*}$:.

$\wedge^{\ell}Varrow$$\Delta\otimes\Delta^{*}$$\phi_{\ell}(<\mathrm{A}, \mathrm{A}, \overline{\mathrm{A}}, \overline{\mathrm{g}_{-}}>)=\sum_{[n]-\mathrm{J}-\mathrm{I}\supseteq \mathrm{K}}\frac{(-1)^{|\mathrm{W}-\mathrm{w}\mathrm{n}\kappa|+|\mathrm{I}|+|\mathrm{K}|}}{2^{(n-|\mathrm{J}|-|\mathrm{I}|)/2}}[B4]$$\otimes[\mathrm{A}4]^{*}$.

This theorem is crucial for us to consider the invariant theory for thegroup Pin(2n) $(O(2n))$ .

36Centralizer algebras for even Spin3.

$\mathrm{I}_{\mathrm{N}\mathrm{V}\mathrm{A}\mathrm{R}1\mathrm{A}\mathrm{N}\mathrm{T}\mathrm{T}\mathrm{H}\mathrm{E}\mathrm{O}\mathrm{R}\mathrm{E}\mathrm{T}\mathrm{I}\mathrm{C}\mathrm{A}\mathrm{L}\mathrm{P}_{\mathrm{R}\mathrm{E}\mathrm{L}\mathrm{I}\mathrm{M}\mathrm{I}\mathrm{N}\mathrm{A}\mathrm{R}\mathrm{I}\mathrm{E}\mathrm{S}}}$

We consider the invariant theory for Pin(2n) $(O(2n))$ . As before,
the First Main Theorem tells us that the invariant symmetric forms
generate the invariants. Since$(\mathrm{v}i, \mathrm{v}j)’ \mathrm{s}$$\mathrm{E}\mathrm{n}\mathrm{d}_{P:n(2n)(\Delta\otimes\otimes^{k}V)=(\Delta^{*}\otimes\Delta\otimes^{2}\otimes^{k}V)^{P}=(\oplus_{=0}\wedge^{l}V\otimes\otimes V)^{O(2n)}}|.n(2n).n^{22k}.$,

it is enough to obtain an explicit base of the invariant polynomials inthe space.

$(\wedge V^{*}\otimes\otimes V^{*})^{O(2n)}(\subset(\otimes^{S}V^{*})^{O(2n)}\subset P(\oplus V)^{O(2n)})rsr+r+s$

We can assume. Those basis elements are multilinear in each variable and has the alternating properties in the first$r+s\equiv \mathrm{O}\mathrm{m}\mathrm{o}\mathrm{d} 2$$r$variables, regarded as the elements in.
As in the case of Spin(2n+1), let $\mathrm{t}=\{t_{1}, t_{2}, \ldots, t_{r}\}(t_{1}<t_{2}<\ldots<$
and $\mathrm{m}=\{m_{1}, \ldots, m_{u}\}$ and $1=\{l_{1}, \ldots, l_{u}\}$ be ordered index sets(or sequences) such that as sets, they are mutually disjoint andsatisfy
the condition
Ul. By, we denote the $u= \frac{s-r}{2}$ pairsof indices.
So the invariant polynomials can be written as sums of the following
polynomials:$P(\oplus V)^{O(2n)}r+s$$t_{r})$$[s]=\mathrm{t}\mathrm{u}\mathrm{m}$$\{\mathrm{m}, 1\}$$\{\mathrm{m}, 1\}=\{\{m_{1}, l_{1}\}, \{m_{2}, l_{2}\}, \ldots, \{m_{u}, l_{u}\}\}$.

$\mathrm{T}_{\mathrm{t},\{\mathrm{m},1\}}=\frac{1}{r!}\sum_{\sigma\in \mathfrak{S}_{r}}\epsilon(\sigma)(\mathrm{X}_{\sigma^{-1}(1),\mathrm{y}_{t_{1}})(\mathrm{X}_{r^{-1}(2),\mathrm{y}_{t_{2}})\ldots(\mathrm{X}_{\sigma^{-1}(r),\mathrm{y}_{t_{r}})\mathrm{x}\prod_{j=1}^{u}(\mathrm{y}_{m_{\dot{f}}},\mathrm{y}_{l_{\mathrm{j}}})}}}$

Here$(j=1,2\ldots, r)$$\mathrm{x}_{j}$$\mathrm{i}\mathrm{n}\otimes^{r+s}V$

tensors inandare the components of the first tensors
are the components of the latter$r$$\mathrm{y}j(j=1,2\ldots, s)$$\otimes^{r\dagger s}V$

Lemma 3.1. Let.

$s$$\mathrm{C}\mathrm{P}_{1}^{\mathrm{k}}=\mathrm{H}\mathrm{o}\mathrm{m}_{P_{\dot{l}}n(2n)}(\Delta\otimes\otimes^{k}V, \Delta\otimes\otimes^{l}V)$, we take only

. If $s=$such that. If $s=$, we take only ’s such that.
Then the above
span linearly the whole space of the.
Moreover if $s=k+l<2n+2$ ,are linearly
independent, $i.e.$ ,$k+l\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} 2$$\mathrm{t}’ s$$|\mathrm{t}|\equiv \mathrm{O}\mathrm{m}\mathrm{o}\mathrm{d} 2$$k+l\equiv 1\mathrm{m}\mathrm{o}\mathrm{d} 2$$\mathrm{t}$$|\mathrm{t}|\equiv 1\mathrm{m}\mathrm{o}\mathrm{d} 2$$\mathrm{T}_{\mathrm{t},\{\mathrm{m},1\}}’ s$$\mathrm{C}\mathrm{P}_{1}^{\mathrm{k}}$$\mathrm{T}_{\mathrm{t},\{\mathrm{m},1\}}$$(\mathrm{t}\mathrm{U} \mathrm{m}\mathrm{u}1=[s])$$\mathrm{C}\mathrm{P}_{1}^{\mathrm{k}}=\bigoplus_{\mathrm{t}\mathrm{u}\mathrm{m}\mathrm{U}1=[s]}\mathbb{C}\mathrm{T}_{\mathrm{t},\{\mathrm{m},1\}}$.

As before we have a natural correspondence between the generalized
Brauer diagramsand the polynomials.
We denotes these elements by adding the suffix’to the diagrams$\mathrm{G}\mathrm{B}_{1}^{\mathrm{k}}$$\mathrm{T}_{\mathrm{t},\{\mathrm{m},1\}}$’$\rho_{-}\mathrm{p}\mathrm{k}$$\mathrm{i}\mathrm{n}\mathrm{v}$37Kazuhiko Koike

4. AREpRESENTATION THEORETIC PARAMETERIZATION
From Theorem 2.1, we can show that-equivariant homomorphism fromand define aas follows.

$\dim(\mathrm{H}\mathrm{o}\mathrm{m}_{Pin(2n)}(\Delta\otimes\wedge^{k}V, \Delta))=$$1$$\Delta\otimes\wedge^{k}V$$\mathrm{P}\mathrm{m}(2\mathrm{n})$$\Delta$

Definition 4.1. If

$k$

to

is even, we define$\mathrm{p}\mathrm{r}_{k}([1]\otimes<-\mathrm{L}, \mathrm{A},\overline{4_{-}},\overline{\mathrm{A}}>)$’$if\mathrm{I}\not\leqq \mathrm{T}if\mathrm{I}\subseteq \mathrm{T}$$=\{_{\epsilon}^{0}$$(\begin{array}{ll}- \mathrm{L} - \mathrm{L} 4\end{array})$If$(-1)^{|\mathrm{W}-\mathrm{w}\mathrm{n}\mathrm{K}|2^{(|\mathrm{I}|+|\mathrm{J}|)/2}}[\mathrm{A}^{\mathrm{g}_{f}]}$is odd, we define$k$$\mathrm{p}\mathrm{r}_{k}$( $[1]\otimes<L,$ -A,$\overline{\mathrm{A}}$,$\overline{4}>$)$if\mathrm{I}\not\subset \mathrm{T}if\mathrm{I}\subseteq \mathrm{T}$.

$=\{_{\epsilon}^{0}$$(\begin{array}{ll}1 arrow \mathrm{I} \mathrm{B}_{f}\end{array})$

Here we put$(-1)^{|\mathrm{w}-\mathrm{W}\cap \mathrm{K}|+|\mathrm{K}|+|\mathrm{J}|2^{(|\mathrm{I}|+|\mathrm{J}|)/2}}[\mathrm{A}\mathrm{K}arrow]$$\mathrm{K}=\mathrm{T}-\mathrm{I}$and$\epsilon$

permutation obtained by arranging$(\begin{array}{ll}3 L arrow \mathrm{K}\end{array})$$arrow \mathrm{T}$into I

denotes the sign of the$arrow \mathrm{K}$Similarly we have-equivariant homomorphism fromfine afollows.

in this order.

$\dim(\mathrm{H}\mathrm{o}\mathrm{m}_{Pin(2n)}\langle\Delta, \Delta\otimes\wedge^{k}V))=1$$\mathrm{P}\mathrm{m}(2\mathrm{n})$$\Delta$toand and de-$\Delta\otimes\wedge^{k}V$asDefinition 4.2. For an even , we define$k$$\mathrm{i}\mathrm{n}\mathrm{j}_{k}([3])=$$(_{-\mathrm{L}}^{\mathrm{T}}arrow$$4$$\mathrm{K}\mathrm{I}\mathrm{I}^{\epsilon}$)($\sum_{\mathrm{J}\mathrm{C}([n]-\mathrm{T})}(-1)^{|\mathrm{w}-\mathrm{w}\mathrm{n}\mathrm{K}|2^{(|\mathrm{J}|+|\mathrm{I}|)/2}}$[A4]$\otimes k!<\mathrm{A},$$\mathrm{A},\overline{\mathrm{A}},\overline{\mathrm{A}}>$$\mathrm{W}\subseteq\overline{(}[n]-\mathrm{I}-\mathrm{J})$$|\mathrm{J}|+|\mathrm{I}|+2|\mathrm{W}|=k$

For an odd , we define$k$$(\begin{array}{ll}- \mathrm{L} - \mathrm{L} arrow \mathrm{K}\end{array})$$\mathrm{i}\mathrm{n}\mathrm{j}_{k}([\sum])=$$\cross$$\sum_{\kappa^{\mathrm{I}\mathrm{C}\mathrm{T}}=\mathrm{T}-\mathrm{I}},\epsilon$($\sum$$\mathrm{J}\mathrm{C}([n]-\mathrm{T})$$\mathrm{W}\subseteq\overline{(}[n]-\mathrm{I}-\mathrm{J})$$|\mathrm{J}|+|\mathrm{I}|+2|\mathrm{W}|=k$$(-1)^{|\mathrm{w}-\mathrm{w}\mathrm{n}\kappa|+|\mathrm{J}|+|\mathrm{K}|2^{(|\mathrm{J}|+|\mathrm{I}|)/2}}$[A4]$\otimes k!<\mathrm{A},$$\mathrm{A},\overline{4},\overline{\iota}>$

)

)

38
Centralizer algebras for even Spin

Using the abovemorphismfrom, we introduce aPin(2n)-equivariant hom0to
as before. Hereasubset of $[k]$ and denotes the
positions in the tensor product. Namely,$\mathrm{p}\mathrm{r}_{\mathrm{P}}$$\Delta\otimes\otimes^{k}V$$\mathrm{p}\mathrm{r}_{\mathrm{A}}$a$\otimes\otimes^{k-p}V$$\mathrm{T}=$$\{t_{1}, t_{2}\ldots, t_{p}\}(t_{1}<t_{2}<\ldots<t_{p})\mathrm{i}\mathrm{s}$$\otimes^{k}V$

Definition 4.3. Let:
be the prOjection map obtained by the composition of the map, $i.e.$ ,and. Hereacts on the alternating tensors sitting in
the positions indexed by I in the space A.
$\Delta\otimes\otimes^{k}Varrow\Delta\otimes\otimes^{k-p}V$$\mathrm{p}\mathrm{r}_{\mathrm{A}}$$\mathrm{A}1\mathrm{t}_{S}$$\mathrm{p}\mathrm{r}_{p}$$\mathrm{p}\mathrm{r}_{1}=\mathrm{p}\mathrm{r}_{p}\mathrm{o}\mathrm{A}1\mathrm{t}_{\mathrm{J}}$$\mathrm{p}\mathrm{r}_{p}$$\otimes\otimes^{k}V$Prom the definition,is the element inand has the alternating property with the index set,$\mathrm{H}\mathrm{o}\mathrm{m}_{Pin(2n)}(\Delta\otimes\otimes^{k}V$$\mathrm{p}\mathrm{r}_{\mathrm{A}}$$T$.

Similarly we define the Pin(2n)-equivariant embedding
as follows.
Definition 4.4. Let$\mathrm{i}\mathrm{n}\mathrm{j}_{1}$:$\Delta\otimes^{k-p}\otimes Varrow\Delta\otimes\otimes^{k}V$a$\otimes 6$$\mathrm{i}\mathrm{n}\mathrm{j}_{1}\in \mathrm{H}\mathrm{o}\mathrm{m}_{P_{\dot{l}}n(2}$be the immersion

obtained by the composition of the map:
and thelinear embedding of the resulting tensors in the positions indexed by .
$\mathrm{i}\mathrm{n}\mathrm{j}_{p}$$\Deltaarrow\Delta$$\otimes\wedge Vp$$\mathrm{A}$Prom the definition,
has the alternating property on the index
set too.
Then we can define representaion-theoretic parameterization of theelements inby the generalized Brauer diagrams
as before.
We fix an element of the diagrams. and letbe its isolated
points in the upper row andbe its isolated points in the lower row.
Then the action represented by the isolated points in the upper row$\mathrm{i}\mathrm{n}\mathrm{j}_{1}$$\mathrm{T}$$\mathrm{C}\mathrm{P}_{1}^{\mathrm{k}}$$\mathrm{G}\mathrm{B}_{1}^{\mathrm{k}}$$\mathrm{G}\mathrm{B}_{1}^{\mathrm{k}}$$\mathrm{T}_{u}$$\mathrm{T}_{\ell}$corresponds to the projection
and the action represented by theisolated points in the lower row corresponds to the immersion.
Namely the total action represented by the isolated points corressponds
to the composition map$\mathrm{p}\mathrm{r}_{\mathrm{T}arrow}$$\mathrm{i}\mathrm{n}\mathrm{j}_{\mathrm{T}}\lrcorner$.

$\Delta\otimes\otimes^{k}Varrow\Delta\otimes\otimes^{k-p}Varrow^{-4}\Delta\otimes\otimes^{k}V\mathrm{p}\mathrm{r}_{\underline{\tau}_{l}}\mathrm{i}\mathrm{n}\mathrm{j}_{\mathrm{T}}$Finally we define the action corresponding to the points which are notisolated just in the same way as those of the ordinary Brauer diagrams.
We denote these elements by adding the suffix ’to the diagramsof.
Whether these elements span linearly the spaceor not, or
whether these elements become abase or not is not clear at present.
We show in the next section that if $k\leqq n$ and $l\leqq n$ , we give theexplicit relations between two parametrizations and that they becomeabase in this case’$\mathrm{r}\mathrm{t}$$\mathrm{G}\mathrm{B}_{1}^{\mathrm{k}}$$\mathrm{C}\mathrm{S}_{1}^{\mathrm{k}}$39Kazuhiko Koike

5. RELATION BETWEEN TWO PARAMETERIZATION
Since the difference between two parametrizations are only in theactions corresponding to the isolated points, we give the relations bebe the isolated points in the upper row
tween them. Let
be the isolated points in the lower row.
and let, deterhomomorphism
We denote theif themined by the invariant polynomial by’or simply by
isolated points are tacitly understood. Here the invariant polynomialwhich we consider in the above is given by$\mathrm{T}_{u}(|\mathrm{T}_{u}|=p)$$\mathrm{T}_{l}(|\mathrm{T}_{l}|=q)$$\Delta\otimes\otimes^{p}Varrow\Delta\otimes\otimes^{q}V$$\psi_{q}^{p}$$\psi_{\vec{\mathrm{T}}}^{\mathrm{T}}\prec$$\sum_{\sigma\in \mathfrak{S}_{r}}\epsilon(\sigma)(\mathrm{X}_{\sigma^{-1}(1),\mathrm{y}_{t_{1}})(\mathrm{X}_{\sigma^{-1}(2),\mathrm{y}_{t_{2}})\ldots(\mathrm{x}_{\sigma^{-1}(r)},\mathrm{y}_{t_{r}})}}$

and we consider this element as the invariant polynomial in the space$(\wedge V)^{*}\otimes\wedge Vp+qp+q$.

The relation of the above two actions are given as follows.
, we haveand
For any$\tau\in \mathfrak{S}_{l}$$\sigma\in \mathfrak{S}_{k}$)$\tau\circ\psi_{4}^{\mathrm{T}}\vec{\tau}\circ\sigma=\psi_{\tau(\mathrm{T})}^{\sigma^{-1}(\mathrm{T}}arrow$. So$\prec$in terms of the
it is enough to give an explicit description for
representation theoretical operators, where $[p]=\{1,2, \ldots,p\}$ and $[q]=$$\{1,2, \ldots, q\}$ .
$\psi_{\Theta}^{[\mathrm{E}|}$

Theorem 5.1. Let $p\leqq n$ and $q\leqq n$ . Then we have
(5.1.1)$\tau\in \mathfrak{S}_{p}\sum_{\in \mathfrak{S}_{q}},\epsilon(\sigma)\epsilon(\tau)\frac{\mathrm{i}\mathrm{n}\mathrm{j}_{\{\sigma([i+1,q])\}}}{(q-i)!}\frac{(_{\sigma([1,i])}^{\tau([1,i])})}{i!}\frac{\mathrm{p}\mathrm{r}_{\{\tau([i+1,p])\}}}{(p-i)!}$$\psi_{\Theta}^{\Theta}=\sum_{\sigma i=0}^{\min(p,q)}$and(5.1.2)$\tau\in \mathfrak{S}_{\mathrm{p}}\sum_{\in \mathfrak{S}_{q}},\epsilon(\sigma)\epsilon(\tau)\frac{\psi_{\{\sigma([i+1,q])\}}^{\{\tau([i+1,p])\}}}{(q-i)!(p-i)!}\otimes\frac{(_{\sigma([1,i])}^{\tau([1,i])})}{i!}$$\circ \mathrm{p}\mathrm{r}_{\Theta}=\sum_{i=0}^{\min(p,q)}(-1)^{i}\sigma$$\mathrm{i}\mathrm{n}\mathrm{j}\Theta$

Here we denote $\sigma([i+1, q])=\{\sigma(i+1), \sigma(i+2), \ldots, \sigma(q)\}$ and$\tau([i+1,p])=\{\tau(i+1), \tau(i+2), \ldots, \tau(p)\}$and$(\begin{array}{l}\tau([1,i])\sigma([1,i])\end{array})$

denotethe partial permutation of the tensor components obtained by sending-th
in the upper row to the-th component inthein the lower row and so on. Also we note that
position in
and inj have the alternating property
in ,occurring
all the indicesrespectively.
So if $k\leqq n$ and $l\leqq n$ , the generalized Brauer diagrams under the representation theoretic parameterization also span the whole space.
and become a linear base of$\Delta\otimes\otimes^{p}V$$\tau(1)$$\sigma(1)$$\Delta\otimes\otimes^{q}V$$\psi$$\mathrm{p}\mathrm{r}$$\mathrm{C}\mathrm{P}_{1}^{\mathrm{k}}$$\mathrm{C}\mathrm{P}_{1}^{\mathrm{k}}$40Centralizer algebras for even Spin

Remark 5.2. The right-hand side of the first for mula 5.1.1 can beconsidered as the composition of the homomorphisms in this order but
the formula 5.1.2 cannot. So we $put\otimes in$ the right-hand side to showthat each summand becomes a homomorphism as a whole.
Example 5.3. We show a few example of the relations of two parameterization. Let us assume that $n\geqq 2$ .
$.\cdot inv=.\cdot$.

$+\cdot$$\mathrm{I}$$.\cdot rt-.\backslash \cdot rt-’$$.\cdot rt+\mathrm{I}$$\mathrm{I}$$rt+\mathrm{I}$$rt-\cross$

$rt$

$rt$

$\mathrm{I}$$\mathrm{I}$$\mathrm{I}$$.\cdot inv=$$.\cdot rt+\mathrm{I}$$rt$

FIGURE 1. The relations of two parameterization
If $p>n$ , we have.
So we need the Pin(2n) isomorphism:
whereis the associater in Lemma 2.3 which is the isomorphism
from
to
and
is aPin(2n) isomorphism from$\det\otimes\wedge^{2n-p}V$
togiven as follows.
$\Delta\otimes\wedge^{p}V\cong\Delta\otimes\wedge^{2n-p}V\otimes\det=\sum_{\dot{l}=0}^{2n-p}[\Delta, (1:)]$$A\otimes r_{p}$$\Delta\otimes\wedge^{p}Varrow\Delta\otimes\wedge^{2n-p}V$$A$$\Delta\otimes\det$$\Delta$$\otimes\wedge^{p}V$$r_{p}$

Lemma 5.4. The sO(2n)-is0m0rphism$:\wedge V\ellarrow 2n\ell\wedge V$$r_{\ell}$is given by follows.
$r_{\ell(<}\mathrm{A}$, where$\mathrm{W}$,$\mathrm{A},\overline{4-}’\tau^{\overline{\mathrm{L}}>)=(-1)^{|\mathrm{I}|}<}\mathrm{A}$$\mathrm{u}\mathrm{w}^{\mathrm{c}}=[n]-\mathrm{J}-\mathrm{I}$

ayzd

,

$\mathrm{u}_{*’\overline{\mathrm{f}^{\mathrm{c}}},6}^{\mathrm{c}}\overline{\mathrm{L}}>$$|\mathrm{J}|+|\mathrm{I}|+2|\mathrm{W}|=\ell$.

The formulas are as follows. As before we define the homomorphismby the composition of the alternating operator Altw and andthe immersion of the alternating tensor to the positions$r_{\underline{2\mathrm{n}-}d}^{1\mathfrak{g}}$$r_{p}$$\frac{[2\mathrm{n}-\mathrm{p}]}{},\cdot$

Theorem 5.5. Let $p>n$ and $q<n$ . Then as a homomorphism fromto, we have$\Delta\otimes\otimes^{p}V$$\Delta\otimes\otimes^{q}V$(5.5.1)$\emptyset_{\Theta}^{\Theta}=(-1)^{q+n}\sum_{\tau\in \mathfrak{S}_{q}}\epsilon(\sigma)\epsilon(\tau)\frac{(_{\tau([1,q])}^{\sigma([1,q])})}{q!}\frac{\mathrm{p}\mathrm{r}_{\{\sigma([q+1,2n-p])\}}}{(2n-p-q)!}(A\otimes r_{\underline{2\mathrm{n}-}\mathrm{g}^{)}}\Theta\sigma\in \mathfrak{S}_{2n-p}\cdot$

We note that from the definition ofequal to $2n$ , $i.e.$ , $p+q\leqq 2n$ .

$\psi_{q}^{p}$, $p+q$ must be less than or,41
Kazuhiko Koike

Let $p<n$ and $q>n$ . Then as a homomorphism from, we have(5.5.2)a$\Delta\otimes\otimes^{p}V$to$\otimes\otimes^{q}V$$\sigma([1,\mathit{1}$$\tau([1,l$$\psi^{[}=(-1\mathfrak{g})^{p+n}\frac{q!}{(2n-q)!}(A\otimes r^{2\mathrm{n}})\Theta\Theta \mathrm{L}\ovalbox{\tt\small REJECT}-\sum_{\sigma\in \mathfrak{S}_{p}}\epsilon(\sigma)\epsilon(\tau)\frac{\mathrm{i}\mathrm{n}\mathrm{j}_{\{\tau([p+1,2n-q])\}}}{(2n-p-q)!}\underline{(}$$p!$$\tau\in \mathfrak{S}_{2n-q}$

6. RELATIONS BETWEEN Pin$(2\mathrm{n})-$EQUIVARIANT HOMOMORPHISMS

In this section we give the relations between the compositions ofand inj and the contractions and the immersion of the invariant forms,
from which we can deduce the multiplication rules of the generalized
Brauer diagrams.

$\mathrm{p}\mathrm{r}$, and the conTheorem 6.1. The following formulas holds for ,and the linear immersion of the invariant eletraction operators
denotes the contraction (with respect to ) of
. Herement
the -th and -th tensor components and$de$ note the immersed positions, namely
and the subindices ofthe first tensor goes to the -th position and the second goes to the j-th$\mathrm{i}\mathrm{n}\mathrm{j}$$\mathrm{p}\mathrm{r}$$\mathrm{C}_{\{i,j\}}$

$i$

$S$

$\mathrm{C}_{\{i,j\}}$$\mathrm{i}\mathrm{d}_{V\{i,j\}}$$\mathrm{i}\mathrm{d}_{\mathrm{y}}=\sum_{i=1}^{n}(u_{i}\otimes u_{\overline{i}}+u_{\overline{i}}\otimes u_{i})$$j$$\mathrm{i}\mathrm{d}_{V\{i,j\}}$$i$position.
(here weto(i) If $p\leqq n$ , as the homomorphisms from
$q+2$
$q+1$
, $p+$,,
positions
in
the
sits
consider the tensor{$q\})$ , we have$\Delta$$\Delta\otimes\otimes^{p}V$$\otimes^{p}V$(6.1.1)$\ldots$$\mathrm{p}\mathrm{r}_{\{[1,q+p]\}}\circ \mathrm{i}\mathrm{n}\mathrm{j}_{\{[1,q]\}}=( 1)^{(p+q\rangle q}(2n-p)_{q\mathrm{P}^{\mathrm{r}}\{[q+1,q+p]\}}$.

Here $(2n-p)_{q}$ denotes the lower factorial, $i.e.$ , for any andnon-negative integer , we define $(x)_{i}=x(x-1)(x-2)\cdots$ $(x-(i-$$1))$ .
The above also holds for $p=0$ and in that case we regards
in the right-hand side as the identity map of .
$p\leqq
n$ , as the homomorphisms(here we(ii) If
from tosits in the positions $\{q+1$ , $q+2$ , $\ldots,p+$consider the tensor$q\})$ , we have
$x$

$i$

$\Delta$$\mathrm{p}\mathrm{r}$$\Delta$$\Delta\otimes\otimes^{p}V$$\otimes^{p}V$(6.1.2)$\mathrm{p}\mathrm{r}_{\{[1,q]\}}\circ \mathrm{i}\mathrm{n}\mathrm{j}_{\{[1,q+p]\}}=(-1)^{(p+q)q}(2n-p)_{q}\mathrm{i}\mathrm{n}\mathrm{j}_{\{[q+1,q+p]\}}$(iii) If $p\leqq n$ and $q\leqq n$ , as the homomorphisms fromwe have(6.1.3)$\Delta$to.

$\Delta\otimes\otimes^{p+q}V$$\mathrm{i}\mathrm{n}\mathrm{j}_{\{[1,q]\}}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{j}_{\{[q+1,q+p]\}}=\sum_{i=0}^{\min(p,q)}(-1)^{qi+()+pq}.2\sum_{\sigma\in \mathfrak{S}_{q}}+1\epsilon(\sigma)\epsilon(\tau)\cross\tau\in \mathfrak{S}_{p}[q]$$\frac{\prod_{u=1}^{i}\mathrm{i}\mathrm{d}_{V\{\sigma(u),\tau(q+u)\}}}{i!}$$\frac{\mathrm{i}\mathrm{n}\mathrm{j}_{\{\sigma([i+1,q]),\tau([q+i+1,q+p])\}}}{(q-i)!(p-i)!}$

.

,

42
Centralizer algebras for even Spin

Here$\mathfrak{S}_{p}[q]$denotes the symmetric groups acting on theacting $\{q+1, q+2, \ldots, q+p\}$ .

(iv) If $p\leqq n$ and $q\leqq n$ , as the homomorphisms from, we have

$p$

char-

$\Delta\otimes\otimes^{p+q}V$

to$\Delta$(6.1.4)$\mathrm{p}\mathrm{r}_{\{[1,q]\}}\mathrm{o}\mathrm{p}\mathrm{r}_{\{[q+1,q+p]\}}=$$\sum_{i=0}^{\min(p,q)}(-1)^{qi+(\begin{array}{l}2\end{array})}\sigma\sum_{\epsilon \mathrm{e}_{q}}\epsilon(\sigma)\epsilon(\tau)\frac{\mathrm{p}\mathrm{r}_{\{\sigma([\dot{l}+1,q]),\tau([q+\dot{l}+1,q+p])\}}}{(q-i)!(p-i)!}\frac{\prod_{u=1}^{\dot{l}}\mathrm{C}_{\{\sigma(u),\tau(q+u)\}}}{i!}$.

$\tau\in \mathfrak{S}_{\mathrm{p}}[q]$(v) If $p\leqq n$ and $q\leqq n$ , as the homomorphisms fromto, (here we consider the tensor
sits in the positions $\{q+1, q+2, \ldots, q+p\}.)$ we have$\Delta\otimes\otimes^{p}V$$\Delta\otimes\otimes^{q}V$$\otimes^{p}V$(6.1.5)$\mathrm{p}\mathrm{r}_{\{[q+1,q+p]\}}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{j}_{[[1,q]\}}=\sum_{\dot{l}=0}^{\min(p,q)}(-1)^{pq}2^{:}\sum_{\sigma\in \mathfrak{S}_{q}}\epsilon(\sigma)\epsilon(\tau)\frac{\mathrm{i}\mathrm{n}\mathrm{j}_{\{\sigma([\dot{|}+1,q])\}}}{(q-i)!}\cross$$\tau\in \mathfrak{S}_{\mathrm{p}}[q]$$\frac{(_{\sigma([1,i])}^{\tau([q+1,q+i])})}{i!}\frac{\mathrm{p}\mathrm{r}_{\{\tau([q+\dot{|}+1,q+p])\}}}{(p-i)!}$Here$(_{\sigma([1,i])}^{\tau([q+1,q+i])})$.

denotes the (partial) permutation which

sends the $\tau(q+u)$ -th component to-th position.
(vi) If $p\geqq t$ and $p-t\leqq n$ and $q\leqq n$ , as the homomorphisms from, (here we consider the tensor
tosits in the positions $\{q+t+1, q+t+2, \ldots, q+p\}.)$ we have$\sigma(u)$$\Delta\otimes\otimes^{p-t}V$a$\otimes^{p-t}V$$\otimes\otimes^{q}V$(6.1.6)$\mathrm{p}\mathrm{r}_{\{[q+1,q+p]\}}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{j}_{\{[1,q+t]\}}=\sum_{\dot{*}=0}^{\min(p-t,q)}(-1)^{pq+t(p+q)}(\sum_{u=0}^{i}(\begin{array}{l}iu\end{array})(2n-p-q+t+i-u)_{t})\cross$$\epsilon(\sigma)\epsilon(\tau)\frac{\mathrm{i}\mathrm{n}\mathrm{j}_{\{\sigma([\dot{l}+1,q])\}}}{(q-i)!}\frac{(\begin{array}{l}\tau([q+t+1,q+t+i])\sigma([1,i])\end{array})}{i!}$$\frac{\mathrm{p}\mathrm{r}_{\{\tau([q+t+\dot{|}+1,q+p])\}}}{(p-t-i)!}$$\sum_{\sigma\in \mathfrak{S}_{q}}$$\tau\in \mathfrak{S}_{p-t}[q+t]$

Herein the parentheses denotes the ordinary binomial c0efficient, If $t=0$ , $(2n-p-q+\mathrm{O}+i-u)_{0}=1$ and we obtain$(\begin{array}{l}iu\end{array})$

6.1.5.

a(vii) If $p\leqq n$ and $q\leqq n$ , as the homomorphisms fromto, (here we consider the tensor
sits in the positions$\Delta\otimes\otimes^{p}V$$\otimes^{q}V$$\otimes\otimes^{q}V$.

43
Kazuhiko Koike$\{p+q+1,p+q+2, \ldots,p+\mathrm{q}\}.)$ we have

(6.1.7)$\prod_{i=1}^{q}\mathrm{C}_{\{i,p+q+i\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{[1,q+p]\}}=\sum_{i=0}^{\min(p,q)}(-1)^{(_{2}^{q})}\sum_{\sigma\in \mathfrak{S}_{q}[q+p]}\epsilon(\sigma)\epsilon(\tau)\frac{\mathrm{i}\mathrm{n}\mathrm{j}_{\{\tau([q+i+1,q+p])\}}}{(p-i)!}$$\tau\in \mathfrak{S}_{\mathrm{p}}[q]$$\frac{(_{\tau([q+1,q+i])}^{\sigma([p+q+1,p+q+i])})}{i!}\frac{\mathrm{p}\mathrm{r}_{\{\sigma([q+p+i+1,2q+p])\}}}{(q-i)!}$.

to(viii) If $p\leqq n$ and $q\leqq n$ , as the homomorphisms fromsits in the posi, (Tiere we consider the tensor, $2p+$
sits in the positions $\{p+q+1,p+q+2$ ,tions$q\}.)$ we have
(6.1.8)$\Delta\otimes\otimes^{q}V$$\otimes^{q}V$$\Delta\otimes\otimes^{p}V$$[q]$$and\otimes^{p}V$$\ldots$$(_{2}^{\mathrm{p}})+pq$$\epsilon(\sigma)\epsilon(\tau)\cross$$\mathrm{p}\mathrm{r}_{\{[1,q+p]\}}\prod_{i=1}^{p}\mathrm{i}\mathrm{d}_{V\{q+i,p+q+i\}}=\sum_{i=0}^{\min(p,q)}(-1)$$\sum_{\sigma\in \mathfrak{S}_{q}}$$\tau\in \mathfrak{S}_{p}[q+p]$$(\begin{array}{l}\sigma([1,i])\tau([p+q+1,p+q+i])\end{array})$$\mathrm{i}\mathrm{n}\mathrm{j}_{\{\tau([p+q+i+1,q+2p])\}}$$\underline{\mathrm{p}\mathrm{r}_{\{\sigma([i+1,q])\}}}$.

$\overline{(p-i)!}\overline{i!}$ $(q-i)$ !
Remark 6.2. If we exchange $2n$ in the above formulas into an in-

determinate $X$ simultaneously, we can define the ‘generic’ centralizer
just as in the ordinary Brauer algebras.
algebra of$\mathrm{C}\mathrm{P}_{\mathrm{k}}$7. EXAMPLES OF pRODUCTS OF THE GENERALIZED BRAUERDIAGRAMS

From the result of the previous section, we can calculate the productof the generalized Brauer diagrams.
As we remarked after the statement of Theorem 6.1, we change $2n$in the formulas of the theorem into an indeterminate $X$ simultaneously
and we write down the relations of the ‘generic’ centralizer algebra of$\mathrm{C}\mathrm{P}_{\mathrm{k}}$.

We summarize other relations between the contractions and the imand inj which follow easily from the definitions. We
mersions andfrom 1to and after
fix the index set of the tensor positions of
we consider the -th and the -th components are
the contractionis allowed if the positions andoccupied by the empty set andare occupied by the empty.
Then we have the followings.
$\mathrm{p}\mathrm{r}$$k$$\otimes^{k}V$

$j$

$i$

$\mathrm{C}_{\{i,j\}}$$i$$\mathrm{i}\mathrm{d}_{V\{i,j\}}$$j$Lemma 7.1.

(i)$\mathrm{C}_{\{s,t\}}\mathrm{i}\mathrm{d}_{V\{i,j\}}=\mathrm{i}\mathrm{d}_{V\{i},{}_{j\}}\mathrm{C}\{s,t\}$if {s,$t\}\cap\{i, j\}=\emptyset$.

44
Centralizer algebras for even Spin

(ii)

if

$\mathrm{C}_{\{s,t\}}\mathrm{i}\mathrm{d}_{V\{tj\}}=(\begin{array}{l}sj\end{array})$Here$s\neq j$.

denotes the (partial) permutation which sends -th component to the -th position.
$(\begin{array}{l}sj\end{array})$

$s$

$j$

(iii)

.

$\mathrm{C}_{\{s,t\}}\mathrm{i}\mathrm{d}_{V\{s,t\}}=(2n)\mathrm{i}\mathrm{d}$(iv) If ,$s$$t\in \mathrm{T}$, we have$\mathrm{C}\{s,t\}\mathrm{i}\mathrm{n}\mathrm{j}_{\mathrm{T}}=0$(v) If ,$s$,$t_{1}$$t_{2}$,$\ldots$,.

$\mathrm{p}\mathrm{r}_{\mathrm{T}}\mathrm{i}\mathrm{d}_{V\{s,t\}}=0$, $t_{r}\in[n]$ are different from each other, we have$(\begin{array}{l}t_{i}s\end{array})$$\mathrm{i}\mathrm{n}\mathrm{j}_{\{t_{1},\ldots,t.,\ldots,t_{r}\}}=\mathrm{i}\mathrm{n}\mathrm{j}_{\{t_{1},\ldots,\dot{s},\ldots,t_{r}\}}.$(vi) If ,$s$$t_{1}$,$t_{2}$,$\ldots$,$(\begin{array}{l}st_{\dot{l}}\end{array})=\mathrm{p}\mathrm{r}_{\{t_{1},\ldots,\dot{s},\ldots,t_{r}\}}.$

$s$

$t$

$t_{1}$,$t_{2}$,$\mathrm{i}\mathrm{d}_{V\{t_{1},t_{2}\}}=\mathrm{i}\mathrm{d}_{V\{s,t_{2}\}}$.

, $t_{r}\in[n]$ are different from each other, we have$\mathrm{p}\mathrm{r}_{\{t_{1},\ldots,t:,\ldots,t_{r}\}}$(vii) If , ,$(\begin{array}{l}t_{1}s\end{array})$$\ldots$,$\mathrm{C}_{\{t_{1},t_{2}\}}$$(\begin{array}{l}st_{1}\end{array})=\mathrm{C}_{\{s,t_{2}\}}$.

, $t_{r}\in[n]$ are different from each other, we have$\mathrm{i}\mathrm{d}_{V\{s,t\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{t_{1},\ldots,t,\ldots,t_{r}\}}:=\mathrm{i}\mathrm{n}\mathrm{j}_{\{t_{1},\ldots,t.,\ldots,t_{r}\}}\mathrm{i}\mathrm{d}_{V\{s,t\}}$and.

$\mathrm{C}_{\{s,t\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{t_{1},\ldots,t,\ldots,t_{r}\}}:=\mathrm{i}\mathrm{n}\mathrm{j}_{\{t_{1},\ldots,t,\ldots,t_{r}}:{}_{\}}\mathrm{C}_{\{s,t\}}$Also we have,$\mathrm{i}\mathrm{d}_{V\{s,t\}}\mathrm{p}\mathrm{r}_{\{t_{1}},\ldots$$t_{:},\ldots,t_{r}\}$:$=\mathrm{p}\mathrm{r}_{\{t_{1},\ldots,t,\ldots,t_{r}\rangle}\mathrm{i}\mathrm{d}_{V\{s,t\}}$and$\mathrm{C}_{\{s,t\}}\mathrm{p}\mathrm{r}_{\{t_{1}},\ldots$,$t_{:},\ldots,t_{r}\}$$=\mathrm{p}\mathrm{r}_{\{t_{1}},\ldots$,$t_{:},\ldots$,$t_{r}{}_{\}}\mathrm{C}_{\{s,t\}}$.

Proof The first three formulas are easy and the fourth and the fifth

formulas follow directly from the alternating property with the indicesofand. The rest are obvious from the definition.
Remark 7.2. If we put, we can deduce easily
the
above
lemma.
These
relations were used to defrom
fine the Birman-Wenzl algebras ( -analog of the Brauer centralizer
in$\mathrm{T}$$\mathrm{p}\mathrm{r}$$\mathrm{i}\mathrm{n}\mathrm{j}$$\square$$e:=\mathrm{i}\mathrm{d}_{V\{:},{}_{:+1\}}\mathrm{C}\{:,:+1\}$$e_{i}e_{i+1}e_{i}=e_{i}$$q$algebras).

Let us show some examples.

Example 7.3. In this example we always assume $n\geqq k$ and exploit the
representation theoretical parameterization of the generalized Brauer
diagrams, so we omit the subscript $rt$ here. We calculate the productin the $k=2$ generalized Brauer diagrams of the introductionof$\mathrm{y}5\mathrm{y}\mathrm{s}$45Kazuhiko Koike:$y_{5}y_{8}=$. .

$-(X-1)$$=$$\mathrm{i}$$(X-1)$–FIGURE 2. The result of the product ofHere$y_{8}=\mathrm{i}\mathrm{n}\mathrm{j}_{\{1},{}_{2\}}\mathrm{C}_{\{1,2\}}$(6.1.2), we haveand$\mathrm{p}\mathrm{r}_{\{1\}}$$(\begin{array}{l}22\end{array})$$\mathrm{i}\mathrm{n}\mathrm{j}_{\{1\}}$. From the for mula

(we put $X$ for $2n.$ ) and$\mathrm{p}\mathrm{r}_{\{1\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,2\}}=(X-1)_{1}\mathrm{i}\mathrm{n}\mathrm{j}_{\{2\}}$the targeting homomorphism is1)$(\begin{array}{l}22\end{array})$$y_{5}=\mathrm{i}\mathrm{n}\mathrm{j}_{\{1\}}$$\mathrm{y}5\mathrm{y}\mathrm{s}$.

$(X-1)\mathrm{i}\mathrm{n}\mathrm{j}_{\{2\}}\mathrm{C}\{1,2\}=(X-$. From the formula (6.1.3), we haveand the final formula is given in Figure 2.
We calculate more complicated general case of Figure 3.

$\mathrm{i}\mathrm{n}\mathrm{j}_{\{1\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{2\}}\mathrm{C}_{\{1,2\}}$$-\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,2\}}-\mathrm{i}\mathrm{d}_{V\{1,2\}}$$=3(X-2)(X-3)$$..\cdot\backslash \cdot..\cdot...\cdot-\overline{\nearrow}......\cdot$$\mathrm{i}\mathrm{n}\mathrm{j}_{\{1\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{2\}}=$.

$.\cdot.\cup^{-}\cdot$$.\sim$$-(X-2)(X-3)(X-4) \sum_{i=1}^{5}\sum_{j=1}^{5}(-1)^{i+j}z_{i}y_{j}$Here$y_{j}$

denotes the upper row and. . . –..

. –..
.. ..
$\cap$.

$z_{i}$

denotes the lower row given as follows.

.
–. –.

$y_{2}$$y_{5}$..

$\mathrm{L}[]\lrcorner$. ..

$y_{3}$.. .

$y_{4}$$\mathrm{L}[]\lrcorner$$z_{1}$..

$z_{3}$$z_{2}$$z_{4}$$z_{5}$FIGURE 3. The result of the product of amore complicated example
In this case we must calculate$\mathrm{i}\mathrm{d}_{V\{3,5\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,2,6,7\}}$$(\begin{array}{l}54\end{array})$$(\begin{array}{l}13\end{array})$$\mathrm{p}\mathrm{r}_{\{1,2,4},{}_{7\}}\mathrm{C}_{\{3,6\rangle}\mathrm{i}\mathrm{d}_{V\{5,7\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,2,4,6\}}$

We note that the inside homomorphism$(\begin{array}{l}54\end{array})$$\mathrm{p}\mathrm{r}_{\{2,3,5},{}_{6\}}\mathrm{C}_{\{4,7\}}$$(\begin{array}{l}\mathrm{l}3\end{array})$$\mathrm{P}\mathrm{r}_{\mathrm{t}^{1,2,4}},{}_{7\}}\mathrm{C}\{3,6\}\mathrm{i}\mathrm{d}_{V\{5,7\}}$$\mathrm{i}\mathrm{n}\mathrm{j}\{1,2,4,6\}$. ( sits in theis an element in
place
inthe lower row.
in
the
upper
and
sits
place
in
the
rout
forth
first
. HereIn general, the inside homomorphism is an element inupper
row to
the
(the
edges
lines
is the number of the verticalfromthe lower row) in the upper diagram and is the number of the vertical$\mathrm{C}\mathrm{P}_{1}^{1}=\mathrm{H}\mathrm{o}\mathrm{m}_{Pin(2n)}(\Delta\otimes V, \Delta\otimes V)$$V$$\mathrm{C}\mathrm{P}_{\mathrm{t}}^{\mathrm{s}}$

$s$

$t$

.

46
Centralizer algebras for even Spinlines in the lower diagram. Each of them corresponds to the indicesin the upper row of the upper diagram and in the lower row of thelower diagram jointed with edges respectively. From Theorem 5.1, thiselement must be a linear combination ofand using the formulasin Theorem 6.1, we can calculate this homomorphism explicitly.
From the formula (6.1.8), we have$\mathrm{G}\mathrm{B}_{\mathrm{t}}^{\mathrm{s}}$$\mathrm{p}\mathrm{r}_{\{1,2,3,4\}}\mathrm{i}\mathrm{d}_{V\{4,5\}}=-\mathrm{i}\mathrm{n}\mathrm{j}_{\{5\}}\mathrm{p}\mathrm{r}_{\{1,2,3\}}-$$(\begin{array}{l}15\end{array})$$\mathrm{p}\mathrm{r}_{\{2,3\}}+(\begin{array}{l}25\end{array})$

If we apply the conjugation of the permutation$\mathrm{p}\mathrm{r}_{\{1,2,4,7\}}\mathrm{i}\mathrm{d}_{V\{5,7\}}=-\mathrm{i}\mathrm{n}\mathrm{j}_{\{5\}}\mathrm{p}\mathrm{r}_{\{1,2,4\}}-$$(\begin{array}{l}35\end{array})$$\mathrm{p}\mathrm{r}_{\{1,3\}}-$$(\begin{array}{l}15\end{array})$$\eta=(\begin{array}{ll}3 44 7\end{array})$$\mathrm{p}\mathrm{r}_{\{1,2\}}$, we have$\mathrm{p}\mathrm{r}_{\{2,4\}}+(\begin{array}{l}25\end{array})$$(\begin{array}{l}45\end{array})$$\mathrm{p}\mathrm{r}_{\{1,4\}}-$$\mathrm{p}\mathrm{r}_{\{1,2\}}$

From the formula (6.1.7), we have$\mathrm{C}_{\{1,5\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,2,3,4\}}=\mathrm{i}\mathrm{n}\mathrm{j}_{\{2,3,4\}}\mathrm{p}\mathrm{r}_{\{5\}}+\mathrm{i}\mathrm{n}\mathrm{j}_{\{3,4\}}$$(\begin{array}{l}52\end{array})-\mathrm{i}\mathrm{n}\mathrm{j}_{\{2,4\}}$

If we apply the conjugation of the permutation$(\begin{array}{l}53\end{array})+\mathrm{i}\mathrm{n}\mathrm{j}_{\{2,3\}}$$(\begin{array}{l}54\end{array})$$\eta=(\begin{array}{lllll}1 2 3 4 56 2 4 1 3\end{array})$.

,

we have$\mathrm{C}_{\{3,6\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,2,4,6\}}=-\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,2,4\}}\mathrm{p}\mathrm{r}_{\{3\}}+\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,4\}}$Since$(\begin{array}{l}32\end{array})-\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,2\}}$$(\begin{array}{l}34\end{array})-\mathrm{i}\mathrm{n}\mathrm{j}_{\{2,4\}}$$(\begin{array}{l}31\end{array})$$\mathrm{p}\mathrm{r}_{\{2,3\}}\mathrm{i}\mathrm{n}\mathrm{j}_{\{1,2\}}=-(X-2)$ $\mathrm{i}\mathrm{n}\mathrm{j}_{\{1\}}\mathrm{p}\mathrm{r}_{\{3\}}-\{(X-1-2+1+1-$$0)+(X-1-2+1+1-1)\}$$\mathrm{P}\mathrm{r}\{2,4\}\mathrm{i}\mathrm{n}\mathrm{j}\{12\}$$(\begin{array}{l}31\end{array})$

from the fomula (6.1.6), we have$=(X-2)\mathrm{i}\mathrm{n}\mathrm{j}\{1\}\mathrm{p}\mathrm{r}_{\{2\}}+(2X-3)$$(\begin{array}{l}21\end{array})$. The calculation

goes on in almost similar rnay and the final result consists of 26 terms
in which only one term contains a vertical line. It is given in the figure

3.

8. DUAL PAIR AND Pin(2n) AND $Sp_{i}n(2n)$ REPRESENTATIONSIn this section we first define the subspace inon whichthe symmetric group of degree and Pin(2n) act as adual pair. Next
we determine the centralizer algebra CSk of Spin(2n) on the spaceand define the subspace on which the symmetric group ofdegree and Spin(2n) act as adual pair.
For the time being, we always assume $n\geqq k$ and we consider thegeneralized Brauer diagrams under the representation theoretic parameterization.
We define the ideals of
as follows.
For the generalized Brauer diagrams and , we consider the insidehomomorphism of the product $yz$ . (See the example 7.3 in the previou$\Delta\otimes\otimes^{k}V$$k$$\Delta\otimes\otimes^{k}V$$k$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$

$y$

$z$

$\mathrm{s}$.

47Kazuhiko Koike

section.) Namely the inside homomorphism is the middle part betweenof and the immersion and inj of .
the contractions and
For ageneralized Brauer diagram we denote by $v(z)$ the numberof the vertical line (the edges from the upper row to the lower row)andin . Then the inside homomorphism is an element in
$y$

$z$

$\mathrm{p}\mathrm{r}$$z$$\mathrm{C}\mathrm{P}_{\mathrm{v}(\mathrm{y})}^{\mathrm{v}(\mathrm{z})}$$z$

are at mostthe vertical lines of the corresponding element in$\min(v(z), v(y))$ .
From Theorem 5.1, the final result consists of the.
elements with at most $\min(v(z), v(y))$ vertical lines in
the linear subspace spanned by all theHence if we denote bybecomes atwo sided
generalized Brauer diagrams with $v(z)\leqq s$ ,, the linear subspace. Also for afixed index set
ideal ofspanned by all the generalized Brauer diagrams in which no verticallines start from the in the upper row becomes aleft ideal and wedenote this left ideal by . Similarly the linear subspace spanned by
all the generalized Brauer diagrams in which no vertical lines end
with the in the lower row becomes aright ideal and we denote this.
right ideal by
At present we don’t know what are the factor algebras of the chainsof the two sided ideal . Only we can say is that the top factor isis the subalgebra. Hereisomorphic to the group algebra
vertical lines, sinceconsisting of the diagrams withofcontains the ordinary Brauer centralizer algebra and the symmetricgroup of degree in natural way. Namely we have$\mathrm{G}\mathrm{B}_{\mathrm{v}(\mathrm{y})}^{\mathrm{v}(\mathrm{z})}$$\mathrm{G}\mathrm{B}_{\mathrm{k}}$$\mathrm{J}_{s}$$z$$\mathfrak{J}_{s}$$\mathrm{T}\subseteq[k]$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$$z$$\mathrm{T}$$L_{\mathrm{T}}$$z$$\mathrm{T}$$R_{\mathrm{T}}$$\mathfrak{J}_{s}$$\mathbb{R}[\mathfrak{S}_{k}]$$\mathbb{R}[\mathfrak{S}_{k}]$$k$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$$k$$\mathrm{C}\mathrm{P}_{\mathrm{k}}=\mathbb{R}[\mathfrak{S}_{k}]\oplus \mathfrak{J}_{k-1}$

Let us define the subspace$T_{k}^{0}$of.

$\Delta\otimes\otimes^{k}V$.

denotes the intersection of all the kernels of
Definition 8.1. Let$\mathrm{C}_{\{i,j\}}(1\leqq
i<j\leqq k)$ and the projections
the contractions( $>0$ and $1\leqq i_{1}<i_{2}<\ldots<i_{r}\leqq k$ ).
$T_{k}^{0}$$\mathrm{p}\mathrm{r}_{\{i_{1},i_{2},\ldots,i_{\mathrm{r}}\}}$$r$acts on thisWe note that any element of the two sided idealPin(2n) subspace. (Actually it becomesis aspace by 0andmodule.) Then we have the following theorem.
an$\mathrm{J}_{k-1}$$T_{k}^{0}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}\mathrm{x}$$\mathfrak{S}_{k}\cross Pin(2n)$Theorem 8.2. Let us assume $n\geqq k$ . Then the symmetric group of. Namely
degree and Pin(2n) becomes a dual pair on the subspace
products
thetensorsum
thedirectdecomposed
intoisof the irof
reducible representations with multiplicity free as follows.
$T_{k}^{0}$$k$$T_{k}^{0}$$(8.2.1)$$\sum$$T_{k}^{0}=$$\lambda$$\lambda_{\mathfrak{S}_{k}}\otimes[\Delta,$:partitions of size$\lambda]_{Spin(2n)}$

Next we move to the case $k>n$ .
$(s=1,2, \ldots, n)$ of
Let us define the subspace.
contains the groups algebra
note that$T_{k,s}^{0}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$.

$k$$\mathbb{R}[\mathfrak{S}_{k}]$$\Delta\otimes\otimes^{k}V$

. We

48
Centralizer algebras for even Spin

Definition 8.3. Let
denote the intersection of all the kernels of
the contractions $\mathrm{C}_{\{:i\}}(1\leqq i<j\leqq k)$ and the projections( $r>0$ and $1\leqq i_{1}<i_{2}<\ldots<i_{r}\leqq k$ ) and the alternating operators$(r>s)$ of degree greater than .
$T_{k,s}^{0}$$\mathrm{p}\mathrm{r}_{\{i_{1},i_{2},\ldots,:_{r}\}}$$\mathrm{A}1\mathrm{t}\{\mathrm{i}_{1},\mathrm{i}_{2},\ldots,\mathrm{i}_{\mathrm{r}}\}$$s$

Then the spacebecome amodule. For, from
Theorem 5.1 and Theorem 5.5, on this space, the homomorphism
corresponding to the isolated points in a generalized Brauer diagrams is 0if. If, then contractions must appear
in the upper row, since the upper and the lower row consist of the same
number of dots. Therefore only in the case that the number of the is0lated points in the upper row is equal to that in the lower row, actson this space non-trivially and in that case, as ahomomorphism, isthe composition of the alternation and the immersion up to ascalar.
Prom the definition,is stable under the action of .
Since
is spanned linearly by the generalized Brauer diagrams
under the invariant theoretic parameterization, the spaceis invariant under the action of. Only the diagrams under the invariant
theoretic parameterization with the same number of the isolated points
in their upper and lower rows and without contractions in their upperrows act on this space non-trivially.
We have the following theorem.
$T_{k,s}^{0}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}\cross Pin(2n)$$T_{k,s}^{0}$$\psi_{\vec{\underline{\mathrm{T}}}}^{\mathrm{T}}4$$|T_{u}|>|T_{\ell}|$$|T_{u}|<|T_{\ell}|$$\psi$$\psi$$T_{k,s}^{0}$$\mathfrak{S}_{k}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$$T_{k,s}^{0}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$Theorem 8.4. Let us assume $k>n$ . Then the symmetric group ofdegree and Pin(2n) becomes a dual pair on the subspace. Exactly
speaking,
is decomposed into the direct sum of the tensor products
theirreducible
representations with multiplicity free as follows.
of$k$$p_{k,s}$$T_{k,s}^{0}$(8.4.1).
$T_{k,s}^{0}=, \sum_{\mathit{0}\lambda:pan\dot{1}t,i(\lambda nsof)\leqq ss\dot{u}ek}\lambda_{\mathfrak{S}_{k}}\otimes[\Delta, \lambda]_{S\dot{\mu}n(2n)}$

We move to the case for Spin(2n).
For Spin(2n) representations, we consider the endomorphism. Here is the associater for and is given by$(A\otimes$$A$$\mathrm{i}\mathrm{d})\in \mathrm{E}\mathrm{n}\mathrm{d}(\Delta\otimes\otimes^{k}V)$$\Delta$

the degree operator $A[i_{1}, i_{2}, i_{3}, \ldots, i_{r}]=(-1)^{r}[i_{1}, i_{2}, i_{3}, \ldots, i_{r}]$ .

Lemma 8.5. The endomorphism
commutes with both of theactions ofand Spin(2n) on the space.
Moreover we have
$A\otimes \mathrm{i}\mathrm{d}$$\Delta\otimes\otimes^{k}V$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$$\mathrm{p}\mathrm{r}_{p}\circ$

(

$A$

c&id)

$=(-1)^{p}A\mathrm{o}\mathrm{p}\mathrm{r}_{p}$and$\mathrm{i}\mathrm{n}\mathrm{j}_{q}\circ A=(-1)^{q}(A$(&id)

Then we first decompose the spaceconstituents under the action of Pin(2n)$\mathrm{i}\mathrm{n}\mathrm{j}_{q}$.
into the irreducible. Since Pin(2n) and$\Delta\otimes\otimes^{k}V$$\cross \mathrm{C}\mathrm{P}_{\mathrm{k}}$49Kazuhiko Koike

are semisimple and act on the space as adual pair, from theWedderburn’s theorem, we have$\mathrm{C}\mathrm{P}_{\mathrm{k}}$.

$\Delta\otimes\otimes^{k}V=\ell$$| \lambda|\leqq k\bigoplus_{(\lambda)\leqq n},[\triangle, \lambda]_{Spin(2n)}\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}$Heresubspacehave$\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}$is the irreducible representation ofas Spin(2n). We consider themodule. Then we$\mathrm{C}\mathrm{P}_{\mathrm{k}}$$\cross \mathrm{C}\mathrm{P}_{\mathrm{k}}$$[\triangle, \lambda]_{Spin(2n)}\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}$.

$[\Delta, \lambda]_{Spin(2n)}\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}=(1/2+\lambda)_{Spin(2n)}^{+}\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}\oplus(1/2+\delta)_{Spin(2n)}\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}$and the space
Since AOid commutes with the action ofis the homogeneous component of the irreducible representation, we haveof$[\triangle, \lambda]_{Spin(2n)}\otimes\lambda \mathrm{c}\mathrm{P}_{\mathrm{k}}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$$\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$$(A\otimes \mathrm{i}\mathrm{d})([\triangle, \lambda]\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}})--[\Delta, \lambda]\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}$.

ineigenspaces of,
Since
modules.
become Spin(2n)eigenspace
eigenspace and
Here
.
in
of
if andappears in
From Theorem 2.1,
appears in. Alsoonly if.
if and only if
is the irreducibleineigenspace ofSo -1
eigenspace ofandrepresentationis the irreducible representationin.
by
We define the extension of the algebra$\mathrm{x}$$A\otimes \mathrm{i}\mathrm{d}$$\pm 1$$(A\otimes \mathrm{i}\mathrm{d})^{2}=\mathrm{i}\mathrm{d}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$$\Delta^{-}\otimes\otimes^{k}V\mathrm{i}\mathrm{s}-1$$\triangle^{+}\otimes\otimes^{k}V\mathrm{i}\mathrm{s}+1$$A\otimes \mathrm{i}\mathrm{d}$$[\triangle, \lambda]\otimes\lambda \mathrm{C}\mathrm{P}_{\mathrm{k}}$$\Delta\otimes\otimes^{k}V$$\triangle^{\pm}\otimes\otimes^{k}V$$(1/2+\lambda)_{Spin(2n)}^{\pm}$$k-|\lambda|\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} 2$$\triangle^{\pm}\otimes\otimes^{k}V$$(1/2+\lambda)_{Spin(2n)}^{\mp}$$k-|\lambda|\equiv 1\mathrm{m}\mathrm{o}\mathrm{d} 2$$[\Delta, \lambda]\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}$$(A\otimes \mathrm{i}\mathrm{d})$$)^{k-|\lambda|}$$($$(-\mathrm{l})^{}$$(1/2+\lambda)_{Spin(2n)}^{+}\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}$$(1/2+\lambda)_{Spin(2n)}\otimes\lambda \mathrm{c}\mathrm{P}_{\mathrm{k}}$$[\Delta, \lambda]\otimes\lambda_{\mathrm{C}\mathrm{P}_{\mathrm{k}}}$$(A\otimes \mathrm{i}\mathrm{d})$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$Definition 8.6. Let
andgenerated by$\mathrm{C}\mathrm{S}_{\mathrm{k}}$$\mathrm{C}\mathrm{P}_{\mathrm{k}}$be the subalgebra of.

$(A\otimes \mathrm{i}\mathrm{d})$$\mathrm{H}\mathrm{o}\mathrm{m}(\triangle\otimes\otimes^{k}V, \triangle\otimes\otimes^{k}V)$$(A\otimes \mathrm{i}\mathrm{d})$Then we have the following theorem,asand Spin(2n) act on the space
Theorem 8.7.
is decomposed into a direct sum
dual pair. Namely the spacewith multiplicity free.
of the irreducible modules of$\triangle\otimes\otimes^{k}V$$a$$\mathrm{C}\mathrm{S}_{\mathrm{k}}$$\triangle\otimes\otimes^{k}V$$\mathrm{C}\mathrm{S}_{\mathrm{k}}\cross Spin(2n)$Then Theorem 8.2 for Spin(2n) is given as follows. Since we havestable and We decompose
in Definition 8.1 isLemma 8.5,
. Thenand denote them by
eigenspaces of
into the
the
case,
have
$|\lambda|=k$
we
this
in
from Theorem 8.2, if we note that
following.
Theorem 8.8. Let us assume $n\geqq k$ . Then the symmetric group of
anddegree and Spin(2n) becomes a dual pair on the subspaces
are decomposed into the direct sum ofrespectively. Namely$(A\otimes \mathrm{i}\mathrm{d})$$T_{k}^{0}$$\pm 1$$T_{k}^{0}$$(A\otimes \mathrm{i}\mathrm{d})$$T_{k}^{0,+}$$k$$T_{k}^{0,-}$$T_{k}^{0,\pm}$$T_{k}^{0,\pm}$.

50
Centralizer algebras for even Spinthe tensor products of the irreducible representations with multiplicity
free as follows.
(8.8.1)$T_{k}^{0,+}= \sum_{\lambda:pan:tions}$$\lambda_{\mathfrak{S}_{k}}\otimes(1/2+\lambda)_{Spin(2n)}^{+}$of sizeand(8.8.1)$T_{k}^{0,-}=$$k$$\sum$$\lambda_{\mathfrak{S}_{\mathrm{k}}}\otimes(1/2+\lambda)_{S\dot{\mu}n(2n)}$

Xipartitions of size.

$k$Theorem 8.4 for Spin(2n) is given as follows.
We assume that $k>n$ .
The subspace $T_{k,s}^{0}(s=1,2, \ldots, n)$ ofin Definition 8.3
is alsostable. We decompose
into theeigenspaces ofand denote them by.
From the same reason as before, we have the following theorem.
Theorem 8.9. Let us assume $k>n$ . Then the symmetric group ofdegree and Spin(2n) becomes a dual pair on the subspaces
andrespectively. Exactly speaking,are decomposed into the direct sum of the tensor products of the irreducible representations withmultiplicity free as follows.
$\Delta\otimes\otimes^{k}V$$(A\otimes \mathrm{i}\mathrm{d})$$(A\otimes \mathrm{i}\mathrm{d})$$T_{k,s}^{0}$$\pm 1$$T_{k,s}^{0,\pm}$$k$$T_{k,s}^{0,\dagger}$$T_{k,s}^{0,-}$$T_{k.s}^{0,\pm}$(8.9.1)$k\lambda_{\mathfrak{S}_{k}}\otimes(1/2+\lambda)_{S\dot{\mu}n(2}^{+}9^{\cdot}$$T_{k,s}^{0,+}= \sum_{\ell(\lambda)\leqq s}\lambda:pan:tionsofs*\cdot ze$(8.9.1)$\lambda_{\mathfrak{S}_{k}}\otimes(1/2+\lambda)_{S\dot{\mu}n(2n)}$$T_{k,s}^{0,-}= \sum_{\ell(\lambda)\leqq s}\lambda panitionsof$

size

$k$

.

51
Kazuhiko Koike

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Trans. Amer. Math. Soc. 313, (1989) pp. 249-273[2] R. Brauer, On algebras which are connected with the semisimple continuous groups, Annals of Mathematics 38, No 4, (1937) pp. 857-872.
57,[3] R. Brauer and H. Weyl, Spinors in n dimensions, Amer. J. Math.
(1935) pp. 425-449[4] K. Koike, Representations of Spinor Groups and the Difference Characters of SO(2n), Adv. Math. 128, (1997) pp. 40-81.
[5] K. Koike, On Representation of the Classical Groups, Amer. Math. Soc.
. 183, (1998) pp. 79-100.
[6] K. Koike, Spin representations and centralizer algebras for Spin(2n+1),
preprint[7] K. Koike, Spin representations and centralizer algebras for Spin(2n),preprint[8] K. Koike and I. Terada, Young-diagrammatic methods for the represen, J. Algebra 107,,
tation theory of the classical groups of type Bn,(1987) pp. 466-511.
[9] H. Weyl, The Classical Groups, their Invariants and Representations,
2nd edition, Princeton Univ. Press, Princeton, N.J., (1946$I\vdash ans$

$C_{n}$$D_{n}$