Certain subrings in classical invariant theory

Certain subrings in classical invariant theory

Nur Hamid , Masashi Kosuda and Manabu Oura

Abstract. We have studied E-polynomials which are combinato- rial analogue of Eisenstein series. In this paper, we apply this ap- proach to classical invariant theory. The corresponding subrings to E-polynomials are investigated.

1. Introduction

Eisenstein series are important in number theory. Under the correspon- dence between combinatorics and modular forms, we have introduced the notion of E-polynomials.

On the other hand, classical invariant theory plays important roles in many branches of mathematics. Igusa [2] discussed invariant theory of bi- nary forms and arithmetic invariants. The connection between the modular forms and projective invariants then was given in [3]. The structure of the graded ring of invariants of binary octavics then was given by Shioda in [8].

In this paper, we construct the analogue theory of Eisenstein series. We apply the notion to classical invariant theory. The computations are done by [1] and [6].

Let m be a positive integer. We take a ground form of degree m f =

X

u

m i

ξ

ξ

.

2010

Primary 13A50; Secondary 13A02, 16W50.

Key words and phrases.

Invariant ring, E-polynomial.

33

Here we mean f is a polynomial of degree m in variables ξ

, ξ

with co- efficients u

where u

are variables. In some parts, we regard f as an element of C[ξ

, ξ

, u

, u

, . . . , u

].

While ξ

, ξ

are transformed according to

(ξ

ξ

) = (ξ

ξ

)A (”contragrediently”),

f changes into a form of the new variables ξ

, ξ

with the coefficients u

, u

, . . . , u

where



 

 

 u

u

.. . u



 

 

 =

A

m



 

 

 u

u

.. . u



 

 

 .

We write this correspondence u

=

A

m

u for short. This gives an irre- ducible representation (A 7→

A

m

) of SL(2, C) of degree m + 1.

We let SL(2, C) operate on C[u] = C[u

, u

, . . . , u

] by the above rep- resentation and consider the invariant subring, say S(2, m):

S(2, m) := {J ∈ C[u] : J(u

) = J(u),

A ∈ SL(2, C)}.

It is known that S(2, m) is of finite type over C. For example, we have S(2, 2) = C[u

u

− u

].

Let M be a graded ring such that each homogeneous part M

of degree d is a finite dimensional vector space over M

= C. We can write M as

M = M

M

.

The dimension formula of M is defined by the formal series X

d=0

(dim M

)t

.

The following formulas are the dimension formulas of S(2, m) for m = 2, 4, 6, 8.

S(2, 2) : X

(dim S

(2, 2))t

= 1 1 − t

, S(2, 4) :

X

(dim S

(2, 4))t

= 1

(1 − t

)(1 − t

) , S(2, 6) :

X

(dim S

(2, 6))t

= 1 + t

(1 − t

)(1 − t

)(1 − t

)(1 − t

) , S(2, 8) :

X

(dim S

(2, 8))t

= 1 + t

+ t

+ t

+ t

Q

i=2

(1 − t

) .

We refer to [7] for the dimension formulas of S(2, 2), S(2, 4), S(2, 6) and to [8] for S(2, 8).

The generators of the rings mentioned are known. For example, S(2, 4) is generated by P and Q whose explicit forms are

P = u

u

− 4u

u

+ 3u

, Q = det



 

u

u

u

u

u

u

u

u

u



  .

The ring S(2, 6) are generated by 5 elements J

, J

, J

, J

, and J

. We write the generators of S(2, 6) in Appendix A. In this paper, we deal with only invariants of even degrees. So we omit J

. We denote by S(2, m)

the even parts of S(2, m).

Let ξ

= ξ

/ξ

. The form ξ

f is a polynomial in ξ

with coefficients in C[u] and can be written as

X

u

m i

ξ

. Let ε

, ε

, . . . , ε

be the roots of the equation

X

u

m

i

ξ

= 0.

We can write

X

ε

= − m

1 u

u

, .. .

X

i1<i2<···<ir

ε

ε

· · · ε

= ( − 1)

m

r u

u

, .. .

ε

ε

· · · ε

= ( − 1)

m

m u

u

.

In order to obtain the useful construction of invariants, we shall interpret the ground form as

f = u

Y

(ξ

− ε

ξ

) .

As usual, we denote by S

the symmetric group of degree n. The following lemma gives a construction of invariants we expected (cf. [2, 9]).

Lemma 1.1. An expression of the form u

X

(ε

− ε

)(ε

− ε

) . . . ,

in which every ε

appears r times in each product and which is symmetric in ε

, ε

, . . . , ε

can be considered as an invariant of degree r.

2. Results

Let g be a positive integer. We start with a ground form of degree 2g + 2 f =

2g+2

X

i=0

u

2g + 2 i

ξ

ξ

= u

Y

i=1

(ξ

− ε

ξ

) .

We would like to concentrate on one type of invariants we shall define now.

We fix the following polynomial

φ

= u

(ε

− ε

)

(ε

− ε

)

. . . (ε

− ε

)

.

We denote by G the symmetric group of degree 2g + 2. The group G acts on the polynomial ring C[ε

, . . . , ε

] as F (. . . , ε

, . . . )

= F(. . . , ε

, . . . ).

Let G

be the stabilizer of φ

, that is, elements of G which preserve φ

. Proposition 2.1. The group G

can be generated by the (g + 1) + 2 elements

(1 2), (3 4), . . . , (2g + 1 2g + 2), (1 3)(2 4), (1 3 5 . . . 2g + 1)(2 4 . . . 2g + 2)

and is isomorphic to C

o S

. In particular, G

does not depend on n.

Proof. The elements given in Proposition 2.1 are in G

. Conversely, since (ε

− ε

)

= (ε

− ε

)

, the first g + 1 elements can be obtained by interchanging of two indexes in each parenthesis. These interchanging are isomorphic to C

. Let ˜ 1, ˜ 2, . . . , g ] + 1 represent (1 2), (3 4), . . . , (g + 1 g + 2), respectively. Then, the additional two generators come from the generators of the set of all permutations of { ˜ 1, ˜ 2, . . . , g ] + 1 } . This set is isomorphic to S

and its generators are (˜ 1 ˜ 2) and (˜ 1 ... g ] + 1) which represent (1 2)(3 4) and (1 3 . . . 2g + 1)(2 4 . . . 2g + 2), respectively.

For simplicity, we denote by K for G

and by κ the cardinality of K\G.

The number κ for g = 1, 2, 3 is 3, 15, 105, respectively.

Set

ψ

= X

K\G∋σ

φ

,

which is actually an element of degree 2n in S(2, 2g + 2) by Lemma 1.1.

We call the polynomial ψ

by an E-polynomial. We shall denote by A

the ring generated by ψ

(n = 1, 2, . . . ) over C. The ring A

is a subring of the invariant ring S(2, 2g + 2).

Theorem 2.2. The ring A

is finitely generated over C. More precisely the elements ψ

, ψ

, . . . , ψ

generate the ring A

.

Proof. Since the second assertion implies the first, we shall show the second.

Let σ

, σ

, . . . , σ

be a set of representatives of K \ G. For each σ

, the

polynomial φ

can be written as

φ

= B

where

B

= (u

(ε

− ε

) (ε

− ε

) . . . (ε

− ε

))

. Thus, the polynomial ψ

has the form

ψ

= B

+ B

+ · · · + B

for n = 1, 2, . . . . By applying the fundamental theorem of symmetric func- tions to our situation, we see that A

is generated by ψ

, ψ

, . . . , ψ

.

The natural question arising from Theorem 2.2 is if we can find the minimal generators of A

. On this point, we have the following theorem.

Theorem 2.3. (1) A

is generated by ψ

, ψ

and coincides with S(2, 4)

. (2) A

is generated by ψ

, ψ

, ψ

, ψ

and coincides with S(2, 6)

. (3) A

is strictly smaller than S(2, 8)

.

Proof. We prove Theorem 2.3 by showing the relationship with the known generators. Starting from g = 1, the polynomials ψ

and ψ

can be ex- pressed in P and Q as

ψ

= 24 · P,

ψ

= 2

· 3 · 11 · P

− 2

· 3

· Q

.

Now we continue for g = 2. By J

, J

, J

, and J

, the polynomials ψ

, ψ

, ψ

, and ψ

can be expressed as

ψ

= − 2

· 3 · 5 · J

,

ψ

= 2

· 3 · 5 · 71 · J

+ 2

· 3

· 5

· J

,

ψ

= −2

· 3

· 5 · 7

· J

− 2

· 3

· 5

· 7J

J

+ 2

· 3 · 5

· 13 · J

, ψ

= 2

· 3

· 5 · 17 · 15287 · J

− 2

· 3

· 5

· 29 · 199 · J

J

+ 2

· 3 · 5

· 37 · 857J

J

− 2

· 3

· 5

· 229J

J

+ 2

· 3

· 5

· 2207 · J

J

− 2

· 3

· 5

· 31 · J

.

For g = 3, the dimension formula of S(2, 8) is X

d=0

(dim S

(2, 8))t

= 1 + t

+ t

+ t

+ t

Q

i=2

(1 − t

)

= 1 + t

+ t

+ 2t

+ 2t

+ 4t

+ 4t

+ 7t

+ · · · . The dimension of S(2, 8) of degree 8 is 7. However, the dimension of A

of degree 8 is at most 5.

For the comparation of the dimension formula, we give an example for g = 2. The dimension formula of S(2, 6) is

1 + t

(1 − t

)(1 − t

)(1 − t

)(1 − t

) , while the dimension formula of A

is

1

(1 − t

)(1 − t

)(1 − t

)(1 − t

) .

In the paper [5], the relation between the ring S(2, 2g +2) and the weight enumerators of some codes was discussed. In Appendix B, we give the relations between the weight enumerators and E-polynomials for g = 1, 2.

Acknowledgment. This work was supported by JSPS KAKENHI Grant Numbers 17K05164, 19K03398. The first named author was also supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. We would like to thank Prof. Hajime Tanaka for his comments.

References

[1] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24(1997), No. 3-4, 235–265.

[2] J. Igusa, Arithmetic variety of moduli for genus two, Ann. of Math., 72(1960), No. 3, 612-649.

[3] J. Igusa, On Siegel modular forms of genus two, Amer. J. Math.,

84(1962), 175-200.

[4] J. Igusa, Modular forms and projective invariants, Amer. J. Math., 89(1967), 817-855.

[5] M. Oura, Observation on the weight enumerators from classical in- variant theory. Comment. Math. Univ. St. Pauli, 54 (2005), No. 1, 1-15.

[6] SageMath, the Sage Mathematics Software System (Version 8.1), The Sage Developers, 2017, https://www.sagemath.org.

[7] I. Schur, Vorlesungen ¨ uber Invariantentheorie, Bearbeitet und heraus- gegeben von Helmut Grunsky, Die Grundlehren der mathematischen Wissenschaften, Band 143 Springer-Verlag, Berlin-New York 1968.

[8] T. Shioda, On the graded ring of invariants of binary octavics, Amer.

J. Math., 89(1967), 1022-1046.

[9] S. Tsuyumine, On Siegel modular forms of degree three, Amer. J.

Math., 108(1986), 755-862.

[10] H. Weyl, The classical groups. Their invariants and representations,

Princeton University Press, Princeton, N.J., 1939.

Appendix A Generators of S(2, 6)

These are the generators of S(2, 6) taken from [7].

J

= u

u

− 6u

u

+ 15u

u

− 10u

,

J

= det



 

 



u

u

u

u

u

u

u

u

u

u

u

u

u

u

u

u



 

 

 ,

J

= det



 

b

b

b

b

b

b

b

b

b



  ,

J

= u

c

− 6u

bc

+ 3u

(ac + 4b

)c − 4u

(3abc + 2b

) + 3u

a(ac + 4b

) − 6u

a

b + u

a

,

where

b

= 6(u

u

− 4u

u

+ 3u

), b

= 3(u

u

− 3u

u

+ 2u

u

), b

= u

u

− 9u

u

+ 8u

, b

= 3(u

u

− 3u

u

+ 2u

u

), b

= 6(u

u

− 4u

u

+ 3u

),

a = 2(u

u

u

− 3u

u

u

+ 2u

u

− u

u

+ 3u

u

u

− u

u

u

− 3u

u

+ 2u

u

),

b = u

u

u

− u

u

u

− u

u

u

− 8u

u

u

+ 9u

u

+ 9u

u

− 17u

u

u

+ 8u

,

c = 2(u

u

u

− u

u

− 3u

u

u

+ 3u

u

u

+ 2u

u

− u

u

u

− 3u

u

+ 2u

u

).

Appendix B Weight Enumerators

We recall coding theory. A code C of length n means a subspace of F

.

The weight wt(x) of x ∈ F

means the number of nonzero x

. The inner

product of two elements x, y ∈ F

is defined by x · y := X

x

y

∈ F

.

The dual code C

of C is defined by the subspace of F

whose elements are orthogonal to every element of C. If C = C

, then we call C self-dual. The code C is called doubly even if the weight of any element in C is equivalent to 0 modulo 4. The weight enumerator of C in genus g is defined by

W

:= W

(x

| a ∈ F

) = X

c1,...,cg∈C

Y

a∈F^g2

x

where

n

(c

, . . . , c

) = |{ i | (c

, . . . , c

) = a }| .

Let ρ e be the combination of the Brou´ e-Enguehard map T h and Igusa’s homomorphism ρ. In other word, we can say

e

ρ(W

) = ρ(T h(W

))

for a code C. We omit the detail of ρ e and only say that ρ e maps the weight enumerators in genus g to the ring S(2, 2g+2). The reader who is interested in the detail of ρ e can refer to [5]. For every code C used here, the expression of ρ(W e

) is taken from [5].

We start with g = 1. The weight enumerators of some codes are related to E-polynomials by the following relations.

e

ρ(W

) = 2

ψ

, e

ρ(W

) = 2

· 11 · ψ

− 2

· 7ψ

.

For g = 2, the relation between the weight enumerators and E-polynomials

are the following.

e

ρ(W

) = 2

ψ

− 3 · 2

ψ

, e

ρ(W

) = 2

3

5

13

31

20129ψ

− 2

513

31

59651ψ

ψ

+ 2

7 · 13

31

809ψ

ψ

− 2

3 · 5

7 · 11 · 31

ψ

ψ

+ 2

3 · 13

31

65287ψ

ψ

+ 2

7 · 13

29 · 31

149ψ

ψ

ψ

− 2

3 · 11ψ

+ 2

3

7 · 13

ψ

, e

ρ(W

d⁺₂₄

) = 2

5

13

31

6323ψ

− 2

13

31

12143ψ

ψ

− 2

3 · 13

31

683ψ

ψ

+ 2

3 · 5

11 · 31

ψ

ψ

+ 2

3

13

31

47 · 379ψ

ψ

− 2

13

31

2089ψ

ψ

ψ

− 2

3 · 11ψ

− 2

3

13

ψ

, e

ρ(W

d⁺₃₂

) = 2

5

13

31

20507ψ

− 2

3

7 · 13

23 · 31

271ψ

ψ

− 2

5

13

23 · 31

227ψ

ψ

+ 2

3

13

15541ψ

ψ

+ 2

3

13

31

4679ψ

ψ

ψ

+ 2

5

13

31

173ψ

ψ

− 2

7 · 13

31

27743ψ

ψ

− 2

13

31

139ψ

ψ

+ 2

3

13

31

2129ψ

ψ

ψ

− 2

13

31

107ψ

ψ

ψ

+ 2

3 · 43ψ

+ 2

13

31

281ψ

ψ

+ 2

3 · 5

13

31

ψ

ψ

,

e ρ(W

d⁺₄₀

) = 31

(2

3

5

13

267941ψ

− 2

3

13

606959 ψ

ψ

− 2

3

13

1877033 ψ

ψ

+ 2

3

13

281

541 ψ

ψ

+ 2

5

13

17 · 4871 ψ

ψ

+ 2

3

13

2207 · 5779 ψ

ψ

ψ

− 2

5 · 13

903827 ψ

ψ

− 2

3

5 · 13

107209 ψ

ψ

− 2

3

5 · 13

17 · 59 · 4957ψ

ψ

ψ

− 2

7 · 13

7187 ψ

ψ

ψ

+ 2

3 · 5 · 13

37 · 205187ψ

ψ

+ 2

3

5 · 13

271919 ψ

ψ

ψ

+ 2

13

17 · 43ψ

ψ

ψ

+ 2

5 · 7 · 13

79319ψ

ψ

ψ

+ 2

3 · 13

181 · 293 ψ

ψ

ψ

− 2

3

19 · 31

ψ

− 2

3

5 · 13

71 · 6719ψ

ψ

− 2

13

17 · 293ψ

ψ

ψ

+ 2

3 · 5

41ψ

).

Nur Hamid

Mathematics Education,

Universitas Nurul Jadid, Probolinggo INDONESIA

e-mail: [email protected] Masashi Kosuda

Yamanashi University JAPAN

e-mail: [email protected] Manabu Oura

Institute of Science and Engineering Kanazawa University

JAPAN

e-mail: [email protected]

(Received May 14, 2020)