数理解析研究所講究録
第 1922 巻 2014 年 60-64

60

ON G-BI-ISOVARIANT EQUIVALENCE BETWEEN-REPRESENTATION SPACES$G$IKUMITSU NAGASAKI AND FUMIHIRO USHITAKIABSTRACT. Let be a compact Lie group. In this paper, we introduce a new equivalentrelation between real -representation spaces, that is, we say that -representation
spaces and $W$ are G-bi-isovariantly equivalent and write as $V=cW$ if there exist-isovariant maps $Varrow W$ and $Warrow V$ . We show that G-bi-isovariant equivalence
between real -representations $W$ with $V^{G}=W^{G}=\{O\}$ implies $DimV=DimW$ ifis finite, or $V\cong W$ if has positive dimension.
$G$

$G$

$G$

$V$

$G$

$G$

$V,$

$G$

$G$

1. INTRODUCTION AND MAIN THEOREM
be a compact
Throughout this paper, all maps are thought to be continuous. LetLie group. Suppose $X$ and are -spaces. Clearly, every -equivariant map : $Xarrow Y$
A -equivariant, wheredenotes the isotropy subgroup of at
satisfies
holds for all $x\in X$ . Inmap : $Xarrow Y$ is called a -isovariant map ifis injective, where $G(x)$ denotes the -orbit
other words, is a -isovariant map if
$G$

$G$

$G$

$Y$

$G$

$G_{x}$$G_{x}\subset G_{\varphi(x)}$$G$$\varphi$$\varphi$$\varphi$

$x.$

$G$

$G_{x}=G_{\varphi(x)}$

$G$

$G$

$\varphi|_{G(x)}$through
In this article, we will consider -isovariant maps between real -representation spaces.
respecand
Let and $W$ be real -representations with the G-fixed point sets
tively. By using Wassermann’s results proved in [6], we can easily show the following
$x.$

$G$

$G$

$V^{G}$$G$$V$$W^{G}$

result.
be a compact solvable LieProposition 1.1 (Isovariant Borsuk-Ulam theorem). Let
group. If there is a -isovariant map : $Varrow W$ , then the Borsuk- Ulam inequality
$G$

$G$

$\varphi$$\dim V/V^{G}\leqq\dim W/W^{G},$that is,$\dim V-\dim V^{G}\leqq\dim W-\dim W^{G}$holds.
Incidently, the reason why Propositon 1.1 is called Isovariant Borsuk-Ulam theorem iswhat it is originated from the Borsuk-Ulam theorem ([1]):2000 Mathematics Subject Classification. Primary $57S17$ ; Secondary $55M20,$ $55M35.$
Key words and phrases. Borsuk-Ulam theorem; Borsuk-Ulam groups; isovariant maps; bi-isovariant
equivalence;transformation groups; finite group action.

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I. NAGASAKI AND F. USHITAKI

Proposition 1.2 (The Borsuk-Ulam theorem). Let
be a cyclic group of order 2.
Assume thatacts on bothandantipodally. If there exists a continuous -map:
, then $m\leqq n$ holds.
$C_{2}$$S^{m}$$C_{2}$$f$$S^{n}$$C_{2}$$S^{m}arrow S^{n}$

It is unknown whether similar statements as Propositon 1.1 hold for any compact Liegroup. The groupis called a Borsuk-Ulam group (BUG) if whenever there is a Gisovariant map : $Varrow W$ , then the Borsuk-Ulam inequality$G$$\varphi$$\dim V/V^{G}\leqq\dim W/W^{G},$that is,$\dim V-\dim V^{G}\leqq\dim W-\dim W^{G}$holds. Wasserman conjectured that all compact Lie groups are BUGs. He gave a suﬃcientcondition called the prime condition for being a BUG. In our previous work [2], we provedthat it is not necessary, that is, we showed there are infinitely many finite groups which
does not satisfy it. For the proof, we introduced a new suﬃcient condition called theM\"obius condition.
In the present work, we introduce a new aspect. Namely, we give insight to the relationship between and $W$ when there exists an isovariant map from not only to $W$
but also $W$ to without the assumption that is a BUG.
$V$

$V$

$V$

$G$

Definition 1.3. Let be a compact Lie group. Let and $W$ be -representations. Wesay thatand $W$ are G-bi-isovariantly equivalent and write as $V_{\vec{-}G}W$ if there exist
$G$

$G$

$V$

$V$

$G$-isovariant maps $Varrow W$ and $Warrow V.$

Clearly G-bi-isovariant equivalence is an equivalent relation, and $V\vec{-}GW$ implies$V\vec{-}HW$ for any subgroup $H.$
Let $S(G)$ be the set of all subgroup of $G,$a real -representation space. We define
the dimension function$V$$DimV:S(G)arrow \mathbb{Z}$

by

$G$

$H\mapsto\dim V^{H}.$Then, we have the following theorem:

Theorem 1.4. Let

be a compact Lie group, and $W$ real -representations such that
$V^{G}=W^{G}=\{O\}$ . Assume $V\vec{-}c$ W. Then, $DimV=DimW$ , that is, $\dim V^{H}=$$\dim W^{H}$ holds
for any $H\in S(G)$ . Moreover, if $\dim G>0$ and is connected, then isisomorphic to $W$ as -representations.
$G$

$V,$

$G$

$G$

$V$

$G$

This article is constructed as follows. In section 2, we give a proof of our theorem whenis finite. In the last section, we explain that isovariant condition is essential in ourresult, and generalize our main theorem.
$G$ON G-BI-ISOVARIANT EQUIVALENCE BETWEEN$G$

62-REPRESENTATION SPACES

2. PROOF OF OUR THEOREM
In this section we prove our theorem when is finite. The non finite case is shown byusin baczyk’s result ([4]), which will be shown in our upcoming paper.
, it holds that
Let be a finite group. For any$G$$g^{}$$H\in \mathcal{S}(G)$$G$.

$\dim V=\frac{1}{|H|}\sum_{g\in H}\chi_{V}(1) , \dim V^{H}=\frac{1}{|H|}\sum_{g\in H}\chi_{V}(g)$Hence,$\dim W-\dim W^{H}-(\dim V-\dim V^{H})=\frac{1}{|H|}\sum_{9\in H}(\chi_{W}(1)-\chi_{W}(g)-\chi_{V}(1)+\chi_{V}(g))$.

Put$h(H)= \sum_{g\in H}(\chi_{W}(1)-\chi_{W}(g)-\chi_{V}(1)+\chi_{V}(g))$.

Then, by [2] we see that$h(H)= \sum_{D\in Cyc1(H)}(\sum_{D\leqq C:cyclic\leqq H}\mu(D, C))h(D)$,where Cyc1 (H) denotes the set of all cyclic subgroups of $H$ , and $\mu()$ is the M\"obiusfunction. Since $D\in Cyc1(H)$ is a BUG and $V\vec{-}cW$ , we have$\dim V-\dim V^{D}=\dim W-\dim W^{D},$that is, $h(D)=0$ by Proposition 1.1. Thus, we have$\dim V-\dim V^{H}=\dim W-\dim W^{H}$for any subgroup $H$ of . Since $V^{G}=W^{G}=\{O\}$ , by choosing$\dim V$ must be equal to $\dim W$ , and consequently $DimV=DimW.$$G$$G$as$H$, we see that

3. REMARKSOur theorem does not hold without the assumption that the maps are isovariant. Waner
gave a necessary and suﬃcient condition for the exixtence of a -map from $S(V)arrow S(W)$with $V\supset W$ , where $S(V)$ and $S(W)$ denote the unit spheres ([5]). By using Waner’s
criterion, we see the following:$G$

Example 3.1. Let $G=C_{pq}$ a cyclic group of order $pq$ where and are distinct prime
be the complex 1-dimensional representation of, letnumbers. For $i=1,$$p$$p,$ $q$

such that$\rho_{i}(g)(z)=\zeta^{i}z$$G$$(T_{i}, \rho_{i})$, where$z\in \mathbb{C}$

$q$

and

$V=T_{1}\oplus T_{p}\oplus T_{q}$$\zeta=\exp\frac{2\pi\sqrt{-1}}{pq}$and. Put$W=T_{p}\oplus T_{q}.$Then, they satisfy Waner’s criterion, thereby, there exist a $G$-map from $S(V)$ to $S(W)$ .

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I. NAGASAKI AND F. USHITAKIAs is stated in Theorem 1.4, if is finite, $DimV=DimW$ holds. Do there exist finitegroups such that $V_{\vec{-}G}W$ imply $V\cong W$ ? At the last of this article, we give insight tothe problem.
Decompose and $W$ into the direct sums of irreducible representations as
$G$

$V$

$V=V_{1}\oplus V_{2}\oplus\cdots\oplus V_{r}$and$W=W_{1}\oplus W_{2}\oplus\cdots\oplus W_{s}.$Then, according to tom Dieck’s book [3], $DimV=DimW$ if and only if $r=s$ and for eachis Galois conjugate to some
, where is a permutation of 1, 2, . . . , , that is,there exists, where $n=LCM\{|9||g\in G\}.$
such that
Thus, we obtain the following:$i,$$V_{i}$$W_{\sigma(i)}$$\psi\in Gal(\mathbb{Q}(\zeta_{n})/\mathbb{Q})$

$r\}$

$\{$

$\sigma$$\psi(\chi_{V_{i}})=\chi_{W_{\sigma(i)}}$

Proposition 3.2. Let
be a finite group, and $W$ real -representations such that
$V^{G}=W^{G}=\{O\}$ . Under the above conditions, if the action ofis trivial
$G$

$G$

$V,$

$Gal(\mathbb{Q}(\zeta_{n})/\mathbb{Q})$$V_{\vec{-}G}W$implies $V\cong W.$

As a corollary, we have :Corollary 3.3. Let
be a finite group. Letand $W$ be real -representation spaces
such that $V^{G}=W^{G}=\{O\}$ . Assume $V\vec{-}G$ W. Then, if, then $V\cong W.$
$G$

$G$

$V$

$\chi_{V}\in \mathbb{Q}$We can illustrate some examples.
be one ofthe following groups. Let and $W$ be real -representation
spaces such that $V^{G}=W^{G}=\{O\}$ . Then, the characters of all real -representations
take the value in . Therefore, $V_{\vec{-}G}W$ implies $V\cong W.$
Example 3.4. Let

$G$

$G$

$V$

$G$

$\mathbb{Q}$: the symmetric group of degree with: the cyclic group of order with $n=2$ , 3, 4, 6.
: the direct product of ’s and, where: the direct product of, where $k\geqq 1.$: the direct product of the quaternion group, where $k\geqq 1.$: the direct product of the diheadral group, where $k\geqq 1.$$\bullet$$\mathfrak{S}_{n}$$\bullet$$C_{n}$$\bullet$$C_{2}^{k}\cross C_{3}^{p}$$\bullet$$C_{4}^{k}$$\bullet$$Q_{8}^{k}$$\bullet$$D_{4}^{k}$$n$$n\in \mathbb{N}.$$n$$C_{2}$$k,$ $\ell\geqq 0.$$C_{3}’ s$$C_{4}’ s$$Q_{8}’ s$$D_{4}’ s$

ON G-BI-ISOVARIANT EQUIVALENCE BETWEEN$G$-REPRESENTATION SPACES

64

REFERENCES[1] K. Borsuk, Drei S\"atze \"uber die -dimensionale Sph\"are, Fund. Math. 20 (1933), 177-190.
[2] I. Nagasaki & F. Ushitaki, New examples of the Borsuk-Ulam groups, RIMS Kokyurokku nessatsu,
B39 (2013), 109-120.
[3] T. tom Dieck, Transformation Groups and Representation Theory, Springer Lecture Notes 766,(1979)[4] P. Traczyk, On the -homotopy equivalence of spheres of representations, Math Z., 161 (1978),
$n$

$G$

257-261[5] S. Waner, A note on the existence of -maps between spheres Proc. Amer. Math. Soc. 99 (1987),
no.1, 179-181[6] A. G. Wasserman, Isovariant maps and the Borsuk-Ulam theorem, Topology Appl. 38 (1991),155$G$

161.
DEPARTMENT OF MATHEMATICS, KYOTO PREFECTURAL UNIVERSITY OFTAKATSUKASA-CHO, TAISHOGUN KITA-KU, KYOTO 603-8334, JAPANMEDICINE, 13 NISH1--mail address: nagasaki@koto.kpu-m.ac.jp (I. Nagasaki)$E$

DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, KYOTO SANGYO UNIVERSITY, KAMIGAMO
MOTOYAMA, KITA-KU, KYOTO 603-8555, JAPAN$E$-mail address: ushitaki@ksuvxO. $kyoto-su$ . ac. jp (F. Ushitaki)