数理解析研究所講究録 1228 巻 2001 年 121-126

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Spherical 5-Designs Obtained fromthe Unitary Group $U_{2m}(2)$Akihiro Munemasa (宗政 昭弘)

1Introduction

The purpose of this talk is to give an infinite series of spherical 5-designs
constructed from the unitary group over the finite field of four elements. Let$G=U_{2m}(2)$ be the unitary group of dimension $2m$ over $GF(4)$ , $V=GF(4)^{2m}$the natural module of . Then acts transitively on the set 0of (maximal)
totally isotropic $m$ -spaces of . This permutation representation (over R)
contains an irreducible representation of dimension $d=(4^{m}+2)/3$ . Then
can embed the set
into the unit spherein the Euclidean space
$G$

$G$

$V$

$S^{d-1}$$\Omega$$\mathrm{o}\mathrm{n}\mathrm{e}\mathbb{R}^{d}$

Theorem 1.

$\Omega \mathrm{c}arrow S^{d-1}\subset \mathbb{R}^{d}$is aspherical 5-design.

The inner product among the vectors of embedded incan be maderational-valued, so one obtains integral lattices after asuitable normalization.
Shimada [5] considered arelated family of lattices, and presented in atalkin January, 2000 at RIMS.
$\Omega$$\mathbb{R}^{d}$2Preliminaries
Aspherical -design$t$$(t \in \mathrm{Z}, t \geq 0)$

is afinite set$\Omega\subset S^{d-1}$

such that$\frac{\int_{S^{d-1}}f(x)dx}{\int_{S^{d-1}}1dx}=\frac{1}{|\Omega|}\sum_{x\in\Omega}f(x)$for all polynomial$f\in \mathbb{R}[X_{1}, \ldots, X_{d}]$of degree at most . Equivalently$\sum_{x,y\in\Omega}Q_{i}(\langle x, y\rangle)=0$$t$$(1\leq i\leq t)$(1)122$Q_{0}(X)=1$ ,$Q_{1}(X)=dX$ ,$\frac{k+1}{d+2k}Q_{k+1}(X)=XQ_{k}(X)-\frac{d+k-3}{d+2k-4}Q_{k-1}(X)$are suitably normalized Gegenbauer polynomials. See $[1, 4]$ for more details
on spherical designs. In what follows we simply say a -design for asphericalt-design.
Examples of spherical designs include the 196, 560 vectors of norm 4in the(a 7-design).
Leech lattice (a 11-design), the 240 roots of the root system
is afinite irreducible subgroup, then every G-0rbit
Moreover, if
is a2-design. SidePnikov [6] showed that there exists afinite groupon
is a7-design. In general,such that every -orbit on
the Molien series of on the space of harmonic polynomials determines forwhich every -orbit on the unit sphere becomes a -design. $[1, 102]$ .
To see that $\Omega\mapsto S^{d-1}(d=(4^{m}+2)/3)$ is a5-design, we shall verify
the condition (1) with $=5$ . We note that the values of inner products
are known to be -2 , $0\leq j\leq m$ (see Table 6.1 (C3) of [3]), andif and only if the dimension of the intersection of and is$m-j$ (recall that , are -dimensional subspaces of ). The number of, where’is given bysuch thatpairs $(x, y)$$t$$E_{8}$$O(d, \mathbb{R})\supset G$$S^{d-1}$$G$$G\subset \mathrm{O}(\mathrm{d},\mathbb{R})$$S^{2^{n}-1}$

$G$

$t$

$G$

$t$

$t$

$($$\langle x, y\rangle$$)^{-j}$$\langle x, y\rangle=(-2)^{j}$

$x$

$x$

$\in\Omega^{2}$$y$$y$

$V$

$m$

$\langle x, y\rangle=(-‘ 2)^{-}$$k_{j}= \prod_{h=1}^{j}\frac{2^{2h-1}(4^{m-h+1}-1)}{4^{h}-1}$$|\Omega|k_{j}$.

With these formulas at our disposal, we can verify (1) for any given values of. However, we shall employ amore general framework to prove Theorem
1.
Acomment on the peculiarity of this embedding can be found in [3,$m$Remark, p.276].

3The -polynomial property for the dual
polar space associated to $U_{2m}(2)$$\mathrm{Q}$As in the previous section, we let be afixed positive integer, and denote by
the set of totally isotropic -spaces in the natural module $V=GF(4)^{2m}$ of$m$$\Omega$$m$

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is called the dual polar space associated to $U_{2m}(2)$ , because
it is acombinatorial dual of the polar space of absolute points and totally
isotropic lines of the projective space $PG(V)$ with aunitary polarity. Then
$U_{2m}(2)$ acts on, and the permutation representation (over R) decomposes
as follows:$U_{2m}(2)$. The set$\Omega$$\Omega$Rn$=V_{0}[perp] V_{1}[perp]\cdots[perp] V_{m}$(2),be the orthogonal projection
where is the trivial module. Let
if necessary, then there
onto . If we rearrange the ordering ofofof degree $i(0\leq i\leq m)$ such thatexists apolynomial$E_{i}\in M|\Omega|(\mathbb{R})$$V_{0}$$\mathbb{R}\Omega$$V_{i}’ \mathrm{s}$$V_{i}$$v_{i}^{*}(X)$$(0\leq i\leq m)$$|\Omega|E_{i}=v_{i}^{*}(|\Omega|E_{1})$,where, if$v_{i}^{*}(X)= \sum_{j=0}^{i}c_{ij}X^{j}$,then$v_{i}^{*}(|\Omega|E_{1})=$

where denotes the entry-wise product. Roughly speaking, the existenceof such polynomials is refered to as the -polynomial property (see [2] forsuch thatdetails). It is known that there exist , ,$0$$\mathrm{Q}$$a_{i}^{*}$$b_{i}^{*}$$c_{i}^{*}\in \mathbb{R}$(3)$Xv_{i}^{*}(X)=c_{i+1}^{*}v_{i+1}^{*}(X)+a_{i}^{*}v_{i}^{*}(X)+b_{i-1}^{*}v_{i-1}^{*}(X)$is asystem of orthogonal polynomials.
andMore generally, one can define acombinatorial structure called an association scheme on which the vector space of real-valued functions on theunderlying set can be decomposed into adirect sum like (2), and one candefine -polynomial property for association schemes. For precise definition,
we refer to [2]. The following theorem reveals arelationship between thethe-polynomial property and spherical designs. Here we denote by
set of unit vectors obtained by normalizing the column vectors of the matri$\{v_{i}^{*}(X)\}$$\Omega$$\mathrm{Q}$$\mathrm{Q}$$E_{1}(\Omega)$$\mathrm{x}$124
Theorem 2. Suppose that is a -polynomial association scheme,, thenis a3-design.
(i) If, then(ii) If moreover,is a4-design.
, thenis a5-design.
(iii) If moreover,$\Omega$$\mathrm{Q}$$E_{1}(\Omega)$$a_{1}^{*}=0$$b_{0}^{*}b_{1}^{*}c_{2}^{*}+2(b_{1}^{*}c_{2}^{*}-b_{0}^{*2}+b_{0}^{*})=0$$a_{2}^{*}=0$$E_{1}(\Omega)$$E_{1}(\Omega)$

If is the dual polar space for $U_{2m}(2)$ , then all hypotheses of the theorem
are satisfied, and becomes a5-design. To check this, we reproduce amore
general formula for these numbers for the dual polar spaces associated with
$U_{2m}(r)$ , whereis aprime power. They can be deduced from the formulasin [2, Section 3.5].
$\Omega$$\Omega$$r$$b_{i}^{*}= \frac{(r^{2m}+r)(r^{2m+2}+(-1)^{i}r^{i+1})}{(r+1)(r^{2m+2}+r^{2i+1})}$$c_{i}^{*}= \frac{r^{i-1}(r^{i}+(-1)^{i-1})(r^{2m}+r)}{(r+1)(r^{2m}+r^{2i-1})}$$a_{i}^{*}=b_{0}^{*}-b_{i}^{*}-c_{i}$,,.

From these formulas, one checks easily that the conditionsof TheOrem 2are satisfied precisely when $r=2$ .
One can find amore general formula describing these numbers for known-and -polynomial association schemes [2, Section 3.5]. Thus, it is natural
to consider the following problem.
$(\mathrm{i})-(\mathrm{i}\mathrm{i}\mathrm{i})$$\mathrm{P}$$\mathrm{Q}$

Problem. Classify P- and -polynomial association scheme
is aspherical -design for t $=4,$ 5,6,$\mathrm{Q}$$E_{1}(\Omega)$$\Omega$such that$t$\ldots.

4Proof of Theorem 2
We use the orthogonality relation of the polyomials$\{v_{i}^{*}(X)\}_{i=0}^{m}$$(i\neq j)$$\sum_{h=0}^{m}k_{h}v_{i}^{*}(\theta_{h}^{*})v_{j}^{*}(\theta_{h}^{*})=0$,

given by(4)

, and $E_{1}())$ has
wherepairs of elementswith inner product. We shall write instead of to simplify thenotation. In view of (1), in order to proveis a -design, it suffices toshow$\theta_{0}^{*}=\dim V_{0}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}E_{1}=b_{0}^{*}$$\theta_{h}^{*}/\theta_{0}^{*}$$|\Omega|k_{h}$$d$$E_{1}(\Omega)$$\sum_{h=0}^{m}k_{h}Q_{i}(\theta_{h}^{*}/d)=0$$\theta_{0}^{*}$$t$$(1\leq i\leq t)$

.

(5)

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Lemma 3. If the polynomials $Q_{s}(X/d)$ (1 s t) are linear combinationsof the polynomials, then $E_{1}(0)$ is ai-design.
$\ovalbox{\tt\small REJECT}$$v\ovalbox{\tt\small REJECT}(X)_{7}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$

Proof. Since$v_{0}^{*}(X)=1$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT} {}_{7}\mathrm{P}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}(X)$, the orthogonality relation (4) implies$\sum_{h=0}^{m}k_{h}v_{i}^{*}(\theta_{h}^{*})=0$$(i>0)$ .

Then the condition (5) is seen to be satisfied.

$\square$

It follows from the definitions that $Q_{1}(X/dX)=X=v_{1}^{*}(X)$ , soalways a1-design. Also, one has$Q_{2}( \frac{X}{d})=\frac{d+2}{2d}(c_{2}^{*}v_{2}^{*}(X)+a_{1}^{*}v_{1}^{*}(X))$

and henceis always a2-design.
To prove part (i) of Theorem 1, we assume$E_{1}(\Omega)$is,$E_{1}(\Omega)$$a_{1}^{*}=0$$XQ_{2}( \frac{X}{d})=\frac{d+2}{2d}c_{2}^{*}Xv_{2}^{*}(X)$, so that

(6)

.

Then
$Q_{3}( \frac{X}{d})=\frac{d+4}{3}(\frac{X}{d}Q_{2}(\frac{X}{d})-(1-\frac{1}{d})Q_{1}(\frac{X}{d}))$$= \frac{d+4}{3d}(XQ_{2}(\frac{Y}{d})-(d-1)Q_{1}(\frac{X}{d}))\wedge$$= \frac{d+4}{3d}(\frac{d+2}{2d}c_{2}^{*}Xv_{2}^{*}(X)-(d-1)v_{1}^{*}(\lrcorner \mathrm{Y}))$$= \frac{(d+4)(d+2)c_{2}^{*}}{6d^{2}}.(c_{3}^{*}v_{3}^{*}(X)+a_{2}^{*}v_{2}^{*}(X))$$+ \frac{(d+4)((d+2)c_{2}^{*}b_{1}^{*}-2d(d-1))}{6d^{2}}v_{1}^{*}(X)$Thus $Q_{3}(X/d)$ is alinear combination ofUnder the assumption of (ii), we have$\mathrm{v}1(\mathrm{X})$,$v_{2}^{*}(X)$,$v_{3}^{*}(X)$$Q_{3}( \frac{X}{d})=\frac{(d+4)(d+2)c_{2}^{*}}{6d^{2}}(c_{3}^{*}v_{3}^{*}(X)+a_{2}^{*}v_{2}^{*}(X))$

.

.

.

(7)

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$Q_{4}( \frac{X}{d})=\frac{d+6}{4}(\frac{X}{d}Q_{3}(\frac{X}{d})-\frac{d}{d+2}Q_{2}(\frac{X}{d}))$$= \frac{(d+6)(d+4)(d+2)c_{2}^{*}}{24d^{3}}(c_{3}^{*}Xv_{3}^{*}(X)+a_{2}^{*}Xv_{2}^{*}(X))-\frac{d+6}{8}c_{2}^{*}v_{2}^{*}(X)$

It follows from (3) that $Q_{4}(X/d)$ is a linear combination of.
,Under the assumption of (iii), we have$v_{3}^{*}(X)$$v_{1}^{*}(X)$,.

$v_{2}^{*}(X)$,$v_{4}^{*}(X)$$Q_{4}( \frac{X}{d})=\frac{(d+6)(d+4)(d+2)c_{2}^{*}}{24d^{3}}c_{3}^{*}Xv_{3}^{*}(X)-\frac{d+6}{8}c_{2}^{*}v_{2}^{*}(X)$which is alinear combination of,is alinear combination of$v_{2}^{*}(X)$$v_{1}^{*}(X)$,$v_{3}^{*}(X)$$v_{2}^{*}(X)$,,$v_{4}^{*}(X)$$v_{3}^{*}(X)$

,,

(8)

by (3). Thus $XQ_{4}(X/d)$,by (3). Since$v_{4}^{*}(X)$$v_{5}^{*}(X)$$Q_{5}( \frac{X}{d})=\frac{d+8}{5}(\frac{X}{d}Q_{4}(\frac{X}{d})-\frac{d+1}{d+4}Q_{3}(\frac{X}{d}))$and $Q_{3}(X/d)$ is ascalar multiple of,,linear combination of
the proof of Theorem 2.
$v_{1}^{*}(X)$$v_{2}^{*}(X)$$v_{3}^{*}(X)$by (7), we see that $Q_{5}(X/d)$ is a,,. This completes$v_{3}^{*}(X)$$v_{4}^{*}(X)$$v_{5}^{*}(X)$

References[1] 坂内英一 坂内悦子, 「球面上の代数的組合せ論」 シュプリンガー. フェアラーク東京, 1999.

[2] E. Bannai and T. Ito, “Algebraic Combinatorics I:Association schemes,”, Menlo Park, Calif., 1984.
Benj$\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{n}/\mathrm{C}\mathrm{u}\mathrm{m}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}$[3] A. E. Brouwer, A. M. Cohen and A. Neumaier, “Distance-RegularGraphs,” Springer, Berlin-Heidelberg, 1989.
[4] P. Delsarte, J.-M. Goethals and J. J. Seidel, Spherical codes and designs,
Geom. Dedicata 6(1977), 363-388.

[5] I. Shimada, Lattices of algebraic cycles on Fermat varieties in positive
characteristics, Proc. London Math. Soc. (3) 82 (2001), 131-172.

[6] V. M. Sidelnikov, Spherical 7-designs in -dimensional Euclidean space,
J. Algebraic Combin. 10 (1999), 279-288$2^{n}$