Prehomogeneous vector spaces obtained from triangle arrangements Takeyoshi Kogisoand Hideto Nakashima

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Prehomogeneous vector spaces obtained from

triangle arrangements

Takeyoshi Kogiso∗and Hideto Nakashima†

Abstract

In this paper, we construct a new series of prehomogeneous vector

spaces from ﬁgures made up of triangles, called triangle arrangements.

Our main theorem states that, under suitable assumptions, we are able

to construct a prehomogeneous vector space obtained from a triangle ar-

rangement by attaching two triangle arrangements corresponding to pre-

homogeneous vector spaces at a vertex. We also give examples of preho-

mogeneous vector spaces obtained from triangle arrangements. Many of

them seem to be new.

Introduction

The theory of prehomogeneous vector spaces, constructed by M. Sato [13] (see

also Sato–Shintani [14], Kimura [8, Introduction]) enables us to construct zeta

functions satisfying functional equations systematically. The key fact is that

basic relative invariants satisfy a local functional equation, that is, the Fourier

transform of a product of complex powers of basic relative invariants is essen-

tially given by a product of complex powers of some polynomials. It is known

that some polynomials which are not basic relative invariants of any prehomoge-

neous vector space satisfy a functional equation (cf. Faraut–Kor´anyi [6], Kogiso–

Sato [9, 10]). Local functional equations are also studied in the ﬁelds of algebraic

geometry and projective geometry (cf. Etingof–Kazhdan–Polishchuk [3]), and

in these ﬁelds, they are related homaloidal polynomials which are homogeneous

polynomials whose gradient-log maps are bi-rational. Many authors including

[1, 2, 4, 5, 7, 11] deal with homaloidal polynomials, and basic relative invari-

ants of regular prehomogeneous vector spaces are recognized as good examples

of homaloidal polynomials (cf. [3]). Therefore, ﬁnding new concrete examples

of regular prehomogeneous vector spaces is important both for the theory of

prehomogeneous vector spaces and algebraic geometry.

In this paper, we construct a new series of prehomogeneous vector spaces

from ﬁgures made up of triangles, called triangle arrangements. Our main the-

orem, Theorem 4.1 states that, under suitable assumptions, we are able to con-

struct a prehomogeneous vector space obtained from a triangle arrangement by

attaching two triangle arrangements corresponding to prehomogeneous vector

∗Department of Mathematics, Josai University, 1-1 Keyakidai, Sakado, Saitama, 350-0295,

Japan. e-mail: kogiso@math.josai.ac.jp

†The Institute of Statistical Mathematics Midori-cho 10-3, Tachikawa, Tokyo 190-8562,

Japan. e-mail: hideto@ism.ac.jp

arXiv:2210.10467v1 [math.RT] 19 Oct 2022

spaces at a vertex. We also give examples of prehomogeneous vector spaces ob-

tained from triangle arrangements in Section 5. Combining results in Section 5

with Theorem 4.1, we are able to construct a lot of prehomogeneous vector

spaces. Many of them seem to be new.

We organize this paper as follows. Section 1 collects a basic tool that we

need later. In particular, a notion of triangle arrangements is introduced. In

Section 2, we view triangulation of convex polygons as triangle arrangements

and consider which triangulation corresponds to a prehomogeneous vector space.

Section 3 is devoted to study a structure of Lie algebras corresponding to triangle

arrangements which have no edge sharing. Our main theorem, Theorem 4.1 is

stated and proved in Section 4. In Section 5, we give four examples of triangle

arrangements which correspond to prehomogeneous vector spaces.

Acknowledgments.

The ﬁst author is supported by the Grant-in-Aid of scientiﬁc research of JSPS

No. 21K03169. The second author was supported by the Grant-in-Aid for JSPS

fellows (2018J00379).

1 Preliminaries

Let V=Cn. We denote the natural representation of GL(V) = GL(n, C) on

Vby ρ. For a given homogeneous polynomial p(x) on V, we introduce a group

G[p] := GL(1) ×G0[p] where

G0[p] := g∈GL(V); pρ(g)x=p(x) for all x∈V.

Then, it is easily veriﬁed that G[p] is an algebraic subgroup of GL(V). By

deﬁnition, we see that p(x) is relatively invariant under the action of G[p]. We

also use the symbol ρfor the action of G[p] on V. In this paper, we work on

the following problem.

Problem 1.1. For which homogeneous polynomial p(x) a triplet (G[p], ρ, V )

admits a structure of a prehomogeneous vector space?

As in [8], the prehomogeneity is an inﬁnitesimal condition so that we shall

describe the condition of a triplet (G[p], ρ, V ) being a prehomogeneous vector

space in terms of Lie algebra. Let g0[p] be the Lie algebra corresponding to

G0[p]. A bilinear form h·|·i on Vis deﬁned to be

hx|yi=t

xy =

i=1

xiyi(x, y ∈V).

Then, we have

g0[p] := {M∈gl(V); hdρ(M)x|∇xp(x)i= 0 for all x∈V},

where dρ is a diﬀerential of ρ. Thus, the Lie algebra g[p] of G[p] is given as

g[p] = gl(1) ˙

+g0[p].(1)

Here, the symbol ˙

+ means a direct sum of vector spaces. By [8, Proposition

2.2], we see that the condition of the triplet (G[p], ρ, V ) being a prehomogeneous

vector space is described by using its Lie algebra g[p] as follows.

Lemma 1.2 (cf. [8, Proposition 2.2]).The triplet (g[p], dρ, V )admits a structure

of prehomogeneous vector space if and only if linear maps A(x): g[p]→V(x∈

V), deﬁned by A(x)M:= dρ(M)x(M∈g[p]) have full generic rank.

In what follows, we concentrate the case of homogeneous polynomials p(x) of

degree three, in particular, those constructed from ﬁgures made up of triangles.

Triangle arrangements are ﬁgures made up of triangles in such a way that

ﬁnite triangles are glued at some vertices or some edges. In what follows, the

symbol Tdenote triangle arrangements. We label number 1,2,3, . . . to vertices

of a triangle arrangement T. We assign variable xito vertex iand, to each

triangle with vertices i, j, k in T, we associate a monomial xixjxk. Then, we

construct a polynomial p(x) from Tby summing up monomials xixjxkwith

respect to each triangle with vertices i, j, k in T. We call p(x) a polynomial with

respect to T, or more simply a polynomial of T.

For brevity, we call a polynomial p(x) is prehomogeneous if the triplet

(g[p], dρ, V ) is a prehomogeneous vector space. Moreover, if p(x) is obtained

from a triangle arrangement, then we also say that Tis prehomogeneous. In

this case, we often write g[T] instead of g[p], where p(x) is a polynomial of T.

Example 1.3. The following ﬁgures are three examples of triangle arrange-

ments.

TATBTC

If we assign a monomial to each grayed triangles, then the corresponding poly-

nomials are given as follows.

pA(x) = x1x4x5+x2x5x6+x3x6x7

pB(x) = x1x2x3+x2x4x5+x3x5x6

pC(x) = x1x5x6+x2x6x7+x2x3x7+x3x4x8

Let Tbe a triangle arrangement with nvertices. Set

T:= {T={i, j, k} ⊂ [n]; a triangle of vertices i, j, k is contained in T},

where [n] := {1,2, . . . , n}. We call Ta hypergraph with respect to T. For each

vertex i∈[n], the set T(i) consists of T∈ T including i, that is,

T(i) := {T∈ T ;i∈T}.

If a vertex isatisﬁes ]T(i) = 1, then iis said to be an isolated vertex. If T

contains two triangles T1,T2such that ](T1∩T2) = 2, then we say that Thas

edge sharing.

For example, TAin Example 1.3 have

T={1,4,5},{2,5,6},{3,6,7},T(5) = {1,4,5},{2,5,6},

and isolated vertices are {1,2,3,4,7}. The triangle arrangement TAdoes not

have an edge sharing, whereas TCdoes.

2 Triangulation of convex polygons

In this section, we view triangulation of convex polygons as triangle arrange-

ments and consider those prehomogeneity. Since prehomogeneity is independent

of the action of GL(V) on p(x), we ﬁrst make a reduction of triangulation of

polygons in order to decrease cases which we consider. Let us explain this

reduction by a concrete example.

The polynomial p(x) associated with Figure 1 (left) is described as

p(x) = x1x2x3+x1x3x4+x1x4x5+x1x5x6.

If we change variables

z2=x2+x4, zi=xi(i= 1,3,4,5,6),

then p(x) transfers to

p(x) = x1x3(x2+x4) + x1x4x5+x1x5x6=z1z2z3+z1z4z5+z1z5z6.

Thus, we can decrease numbers of monomials. In terms of ﬁgures, we focus on

the vertex 2 and a triangle 123, and vanish a triangle sharing edges with the

triangle 123. This polynomial can be further transferred by changing variables

w6=z6+z4, wi=zi(i= 1,2,3,4,5)

p(z) = z1z2z3+z1z5(z4+z6) = w1w2w3+w1w5w6,

and we ﬁnally get a polynomial consisting of two monomials. Along this re-

duction, triangles having the vertex 4 disappear. Since the original triangu-

lation have 6 vertex, we should calculate for 5-variable polynomial p(w) on

6-dimensional vector space C6.

In Figure 4, we exhibit triangle arrangements obtained by reduction of trian-

gulation of n-polygons up to n= 10. We can see prehomogeneity at the top of

each ﬁgure. Its proofs are left to Section 5. A black circle •in ﬁgures indicates

a vertex which does not appear as a vertex of triangles, like the vertex 4 in the

above example. For simplicity, triangle arrangements with a black circle are

also called just triangle arrangements. Note that, if a triangle arrangement Tis

a triangle arrangement T0with a black circle (T0does not have a black circle),

then the corresponding Lie algebras g[T] and g[T0] are related as

g[T] = M=M00

txm;M0∈g[T0],x∈C]T0

, m ∈C.

Figure 1: Reduction of a hexagon triangulation

n6 7 8 9 10 11 12 13 14 15 16 17

(A) 3 4 12 27 82 228 733 2282 7528 24834 83898 285357

(B) 3 2 7 7 26 37 137 298 993 2726 8749 26446

Table 1: The row of (A) indicates numbers of triangulation of n-polygons up to

rotations and reﬂections, (B) those of reduced triangle arrangements and (C)

those of prehomogeneous vector spaces.

In particular, the prehomogeneity of Tis the same as that of T0.

We see from Figure 4 that triangle arrangements obtained by reduction of

triangulation of n-gons may unconnected. We shall show in Proposition 5.2 that

they cannot be prehomogeneous unless one of the connected components is a

black circle discussed in the previous paragraph.

Table 1 includes numbers of (A) triangulation of n-polygons under rotations

and reﬂections, (B) those of reduced triangle arrangements and (C) those of

prehomogeneous vector spaces.

3 Structure of g[p]without edge sharing

Let Tbe a triangle arrangement with nvertices and p(x) the corresponding

polynomial. Suppose that Thas no edge sharing. In this section, we investigate

a structure of g[p], which will be needed to prove our main theorem. By (1),

it is enough to calculate hdρ(M)x|∇xp(x)i= 0. By deﬁnition, p(x) can be

described by using the hyper graph Tassociated with Tas

p(x) = X

{i,j,k}∈T

xixjxk.

Thus, we have

hdρ(M)x|∇xp(x)i=

i=1

a=1

Miaxa·X

{i,j,k}∈T (i)

xjxk

i=1

a=1 X

{i,j,k}∈T (i)

Miaxaxjxk.

(2)

We ﬁrst exhibit a calculation of a Lie algebra of a concrete polynomial.

Example 3.1. Let p(x) be a homogeneous polynomial whose hyper graph is

given as T={{1,2,3},{1,4,5}}, that is,

p(x) = x1x2x3+x1x4x5.

In this case, we have

hdρ(M)x|∇xp(x)i

=M11x1(x2x3+x4x5) + M12x2(x2x3+x4x5) + M13x3(x2x3+x4x5)

+M14x4(x2x3+x4x5) + M15x5(x2x3+x4x5)

+M21x1·x1x3+M22x2·x1x3+M23x3·x1x3+M24x4·x1x3+M25x5·x1x3

+M31x1·x1x2+M32x2·x1x2+M33x3·x1x2+M34x4·x1x2+M35x5·x1x2

+M41x1·x1x5+M42x2·x1x5+M43x3·x1x5+M44x4·x1x5+M45x5·x1x5

+M51x1·x1x4+M52x2·x1x4+M53x3·x1x4+M54x4·x1x4+M55x5·x1x4

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PrehomogeneousvectorspacesobtainedfromtrianglearrangementsTakeyoshiKogiso*andHidetoNakashimaAbstractInthispaper,weconstructanewseriesofprehomogeneousvectorspacesfromguresmadeupoftriangles,calledtrianglearrangements.Ourmaintheoremstatesthat,undersuitableassumptions,weareabletoconstructaprehomogeneo...

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Prehomogeneous vector spaces obtained from triangle arrangements Takeyoshi Kogisoand Hideto Nakashima

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