研究集会「組合せ論的表現論とその周辺」
複素単純LIE群の外部自己同型の重複度 1 の分岐則につぃて$,$
MULTIPLICITY-FREE BRANCHING RULES
FOR OUTER AUTOMORPHISMS OF SIMPLE LIE ALGEBRAS
有川英寿 ALIKAWA HIDEHIS
We
announce
the paper [2].1. INTRODUCTION
1.1. Let $\mathfrak{g}$ be acomplex semisimple Lie algebra, and $\mathfrak{g}’$ be areductive Lie
subalgebra of $\mathfrak{g}$. The restriction $\pi|_{\mathfrak{g}’}$ of
a
irreducible representation $\pi$ of gneed not be irreducible.
The irreducible decompsiton of $\mathfrak{g}’$
$\pi|_{9’}=$ $\oplus$ $c_{\pi}^{\mu}\mu$
$\mu \mathrm{i}\mathrm{s}$ irreduciblerepresentationof$\mathfrak{g}’$ is called branching rule.
Problem 1.
Exmaple 1. There are well-known branching rules.
(1) classical rule. Set $\mathfrak{g}$ $=\epsilon \mathrm{I}_{n+1}$, $\mathfrak{g}’=\epsilon \mathfrak{l}_{n}$, then irreducible
represen-tations of gare indexed by $\lambda=$ $(\lambda_{1}, \ldots, \lambda_{n})$ with $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq$
$\lambda_{n}\geq 0$. In this case, we have the complete answer to Problem 1.
We have
$\lambda|_{\mathfrak{g}’}=\oplus\lambda_{1}\geq\mu_{1}\geq\lambda_{2}\geq\cdots\geq\mu_{n-1}\geq\lambda_{n}\mu$
.
In particular,
we
have $c_{\pi}^{\mu}\leq 1$.Date: Nov 28, 2002
数理解析研究所講究録 1310 巻 2003 年 178-185
179
$\mathrm{f}\mathrm{i}l||\mathrm{m}*\ovalbox{\tt\small REJECT}$
(2) highest weight theory. Let $\mathfrak{g}’$ be aCartan subalgebra of
$\mathfrak{g}$. The
answer of Problem 1is the Cartan-Weyl’s highest weight theory. The
branching rule is decomposition of weight spaces. The description of
$c_{\pi}^{\mu}$ is Kostant’s formula.
(3) tensor product Let $\mathfrak{g}$ $=\mathfrak{g}’\cross \mathfrak{g}’$ and
$\mathfrak{g}’$ diagonal in
$\mathrm{g}$. The irreducible
representationof
9is
given by $\pi=\sigma \mathbb{H}\tau$where$\sigma$ and $\tau$are
irreduciblerepresentations of $\mathfrak{g}’$. The branching rule $\pi|_{\mathfrak{g}’}=\pi\otimes\sigma$ is the tensor
product, which
causes
Littlewood-Richardson rule.Remark 2. Koike-Terada [9] gave general formulas of $GL(n)$ to SO(n)
or
$GL(2n)$ to $Sp(n)$ by using the universal characters.
Problem 2.
This branching rule is called multiplicity-free.
Remark 3. It is difficalt to get the weight mutiplicity-free representations,
though we have the Kostant’s general formula. Similarly, we do not have
the classification of mutiplicity-free branching rules, though we have the
general formula Koike-Terada’s algorigthm.
Exmaple 4. We have some
answers
to the emaples in Example 1.(1) always
(2) few
(3) few (Multiplicity-free tensor products are called Clebsch-Gordan’s
rule, which
are
cllasified by Stembridge [13].Remark 5. Kobayashirecently obtainedan abstract theorem of
multiplicity-free branching rules for both infinite and finite dimensional representations
for ageneral symmetric pair $(G, G’)[6][8]$.
Okada
uses
newcombinatorialformulason minors due toIshikawa-Wakayama[5]to obtain explicit branching rules [11].
We want
anew
technique in getting many branching rules.In the paper [2],
we
get the many examples of multiplicity-free branchingrules, which
we
intoduce in this proceeding$\theta_{\grave{\mathrm{J}}}\mathbb{R}\ovalbox{\tt\small REJECT}^{1}\mathrm{J}$
2.
SETTING.
Let
9be
acomplex simple Lie algebra, abe aDynkin diagramaut0-morphism of $\mathfrak{g}$, and $\mathfrak{g}’=\mathfrak{g}^{\sigma}:=\{X\in \mathfrak{g}|\sigma X=X\}$. We choose
acr-stable Cartan subalgebra [$)$ in $\mathrm{g}$ such that $\mathfrak{h}^{\sigma}:=\{X\in \mathfrak{h}|\sigma X=X\}$ is aCartan
subalgebra of $\mathfrak{g}^{\sigma}$. We shall
use
thesame
notation$\sigma$ to denote the natural
action
on
[$)$, and also $\mathfrak{h}^{*}$. Let $\triangle\equiv\Delta(\mathfrak{g}, \mathfrak{h})$ be the rootsystem of
9with
respect to the
Cartan
subalgebra $\mathfrak{h}$, and $\triangle^{+}$ be positive roots.Case 1
(A2, Ai) that is $(\epsilon 1(3,\mathbb{C}),\epsilon 1(2,\mathbb{C}))$.
$\square _{2}^{1}$
$A_{2}$ $\mathrm{O}^{1}$ $A_{1}$
Case 2
$(A_{2m-1},C_{m})$ (m $\geq 2)$ that is $(\epsilon 1(2m,\mathbb{C}),\epsilon \mathfrak{p}(m,\mathbb{C}))$.
1 2 m-1
$m$ $A_{2m-1}$
$c_{m}$
Case 3
$(A_{2m}, B_{m})$ $(m\geq 2)$ that is $(\epsilon 1(2m+1, \mathbb{C})$,so$(2m+1, \mathbb{C}))$. Case 4
$(D_{m}, B_{m-1})$ $(m\geq 4)$ that is $(\epsilon \mathrm{o}(2m, \mathbb{C}),$$\epsilon 0$$(2m-1, \mathbb{C}))$.
Case 5
$(E_{6}, F_{4})$.
181
$k\mathrm{I}\mathrm{I}|^{rightarrow}\#\ovalbox{\tt\small REJECT}$ 1 2 m–1 $m$ $A_{2m}$ $B_{m}$ $m-1$ $D_{m}$ $m$ $B_{m-1}$ 2 1 $E_{6}$ $F_{4}$ Case 6 $(D_{4},G_{2})$. 2 $D_{4}$ $\mapsto 12$ $G_{2}$Remark 6. We remark that there is adetailed study of $(\mathfrak{g}, \mathfrak{g}^{\sigma})$ when
9is
ageneralized Kac-Moody Lie algebra by Fuchs-Schellekens-Schweigert [3]
and Fuchs-Ray-Schweigert [4].
Remark
7.
Only in Case 6, the order of $\sigma$ is three. The pairs $(\mathfrak{g}, \mathfrak{g}^{\sigma})$ inCases 1-5
are
called symmetric pairs$\theta_{\mathrm{J}}^{\backslash }\mathbb{R}\ovalbox{\tt\small REJECT}|\rfloor$
3. MAIN RESULTS
We denote by $X_{n}(\lambda)$ the irreducible finite dimensional representation of
acomplex simple Lie algebra of type $X_{n}(X=A, B, C, D, E, F, G)$ with a
highest weight $\lambda$, and by
$X_{n}(\lambda)|_{\mathrm{Y}_{n}}$, the restriction to acomplex Lie algebra
$\mathfrak{g}’$ of type $Y_{n’}$.
Let $\{\varpi_{j}\}_{j=1}^{n}$ be fundamental weights, with respect to afixed simple
sys-tem $\{\alpha_{j}\}_{j=1}^{n}$ of acomplex Lie algebra of type $X_{n}$ or $\mathrm{Y}_{n’}$, which
are
labeledin the previous subsection.
Theorem 1. For $k\in \mathrm{N}_{f}$
(2A) $A_{2m-1}(k\varpi_{1})|_{C_{m}}=A_{2m-1}(k\varpi_{2m-1})|_{C_{m}}=C_{m}(k\varpi_{1})$ (m $\geq 2)$
(4A) $D_{m}(k\varpi_{m-1})|_{B_{m-1}}=D_{m}(k\varpi_{m})|_{B_{m-1}}=B_{m-1}(k\varpi_{m-1})$ (m $\geq 4)$
Theorem 2. For $k$, $l$ $\in \mathrm{N}$,
(IB) $A_{2}(k\varpi_{1})|_{A_{1}}=A_{2}(k\varpi_{2})|_{A_{1}}=\oplus^{k}A_{1}(s\varpi_{1})s=0$
(2B) $A_{2m-1}(k\varpi_{1}+l\varpi_{2})|_{C_{m}}=A_{2m-1}(k\varpi_{2m-1}+l\varpi_{2m-2})|_{C_{m}}=$
$\oplus^{l}C_{m}(k\varpi_{1}+s\varpi_{2})s=0$ (m $\geq 3)$
(3B)
$A_{2m}(k\varpi_{1})|_{B_{m}}=A_{2m}(k\varpi_{2m})|_{B_{m}}=\oplus_{s\leq k}B_{m}(s\varpi_{1})s\equiv k\mathrm{m}\mathrm{o}\mathrm{d} 20\leq$
(m $\geq 2)$
(4B) $D_{m}(k\varpi_{1})|_{B_{m-1}}=\oplus^{k}B_{m-1}(s\varpi_{1})s=0$ (m $\geq 4)$
(5B) $E_{6}(k\varpi_{1})|_{F_{4}}=E_{6}(k\varpi_{6})|_{F_{4}}=\oplus^{k}F_{4}(s\varpi_{4})s=0^{\cdot}$
$\mathrm{f}\mathrm{i}\mathrm{I}||\vec{\Re}\ovalbox{\tt\small REJECT}$
(6B) $D_{4}(k\varpi_{1})|_{G_{4}}=D_{4}(k\varpi_{3})|_{G_{4}}=D_{4}(k\varpi_{4})|c_{4}=\oplus^{k}G_{2}(s\varpi_{2})s=0$
Remark 8. Some of these branching rules are new. One can prove
some
of them in several ways by using Borel-Weil theory, Gelfand-Tsetlin basis,
formulas of minors, and
so on
(See, for example, [7], [14], [12], [11], [10]),4.
SKETCH
OF $\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{O}\mathrm{F}}$We write down the sketch of proof of the theorems by using Weyls
character formula and denominator formula (See [2]).
Let $X_{n}(\lambda)$ be the representation which appears in left hand side of
The-orems
1and 2.Let char$X_{n}(\lambda)$ be the character of $X_{n}(\lambda)$.
We write $\rho_{X_{n}}$, $d_{X_{n}}$, $\triangle_{X_{n}}^{+}$, $W_{X_{n}}$ as halfsum of positive roots, Weyl
denomi-nator, positive roots ofcomplex simple Lie algebraoftype $X_{n}$, Weyl group,
respectively.
By Weyl’s character formula,
char$X_{n}( \lambda)=d_{X_{n}}^{-1}\sum_{w\in W_{Xn}}\epsilon(w)e^{w(\lambda+\rho \mathrm{x}_{n})}$
(We set $W_{X_{n}}(\lambda):=\{w \in W_{X_{n}}|w\lambda=\lambda\}$ and $W_{X_{n}}^{\lambda}$ minimal representatives
of $W_{X_{n}}/W_{X_{n}}(\lambda).)$
$=d_{X_{n}}^{-1} \sum_{w_{1}\in W_{X_{n}}^{\lambda}}(\sum_{w_{2}\in W_{Xn}(\lambda)}\epsilon(w_{1}w_{2})e^{w_{1}w_{2}\lambda+w_{1}w_{2}\rho \mathrm{x}_{n)}}$
$=d_{X_{n}}^{-1} \sum_{w_{1}\in W_{\chi_{n}}^{\lambda}}(\epsilon(w_{1})e^{w_{1}\lambda}(\sum_{w_{2}\in W\mathrm{x}_{n}(\lambda)}\epsilon(w_{2})e^{w_{1}w_{2}\rho x_{n)}})$
$\theta_{\grave{\mathrm{J}}}\mathbb{R}^{\mathrm{B}_{\backslash }1},\mathrm{J}$
By the denominator formula for $W_{X_{n}}(\lambda)$,
$\sum_{w_{2}\in W_{X_{n}}(\lambda)}\epsilon(w_{2})e^{w_{2}\rho x_{n}}=e^{\rho x_{n}}\prod_{\alpha\in\Delta_{X_{n}}^{+}(\lambda)}(1-e^{-\alpha})$.
Applying $w_{1}\in W_{X_{n}}^{\lambda}$,
$\sum_{w_{2}\in W_{X_{n}}(\lambda)}\epsilon(w_{2})e^{w_{1}w_{2}\rho x_{n}}=e^{w_{1}\rho x_{n}}\prod_{\alpha\in\Delta_{X_{\hslash}}^{+}(\lambda)}(1-e^{-w_{1}\alpha})$.
Then,
(X)
char$X_{n}( \lambda)=d_{X_{n}}^{-1}\sum_{w_{1}\in W_{X_{n}}^{\lambda}}(\epsilon(w_{1})e^{w_{1}(\lambda)}(e^{w_{1}\rho x_{n}}\prod_{\alpha\in\Delta_{x_{n}}^{+}(\lambda)}(1-e^{-w_{1}\alpha})))$
In the
same
way,we
calculate char$\mathrm{Y}_{n’}(\lambda’)$. $(\lambda’=\lambda|_{\mathfrak{h}^{\sigma}})$(Y) char
$\mathrm{Y}_{n’}(\lambda’)=d_{\mathrm{Y}_{n}}^{-1},\sum_{w_{1}\in W_{\mathrm{Y}_{n}}^{\lambda}},,$
$( \epsilon(w_{1})e^{w_{1}(\lambda’)}(e^{w_{1}\rho_{\mathrm{Y}_{n’}}}\prod_{\alpha\in\Delta_{\mathrm{Y}_{n}}^{+}(\lambda)},’(1-e^{-w_{1}\alpha})))$
Lemma 3. $W_{X_{n}}^{\lambda}$ and $W_{\mathrm{Y}_{n}}^{\lambda’}$,
are
“equal”.In explicit, in the situation
of
Theorem 1, $W_{X_{n}}^{\lambda}$ and $W_{\mathrm{Y}_{n}}^{\lambda’}$,are
equal. In thesituation
of
Theorem 2, $W_{X_{n}}^{\lambda}\backslash W_{\mathrm{Y}_{n}}^{\lambda’}$, can be characterized by $w\varpi|_{\mathrm{b}^{\sigma}}=0$.Remark9. This lemma may be mysterious, because $W_{Y_{n}}$, is much smaller than $W_{X_{n}}$. Lemma 4. The summands of(X) and (Y) are “equal”.
In explicit, in the situation of Theorem 1, the summands are equal. In the situation of Theorem 2, the difference of each summands is only one term.
Wecan prove the theoremsbyusing themysterious lemmas, in particularLemma 3. We prove these lemmasby case-by-case calculation, then we donot understandwhy Lemma3
is true.
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有川英寿
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京都大学数理解析研究所, RIMS, Kyoto UNIVERSITY, KYOTO 606-8502, JAPAN
$E$-mail address: alleyOkurims. kyoto-u.ac.jp
