Certain subrings in classical invariant theory

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84(1962), 175-200.

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Nur Hamid, Masashi Kosuda and Manabu Oura

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89(1967), 817-855.

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Sage Developers, 2017, https://www.sagemath.org.

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¨ber Invariantentheorie, Bearbeitet und herausgegeben von Helmut Grunsky, Die Grundlehren der mathematischen

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

Certain subrings in classical invariant theory

Appendix A

41

Generators of S(2, 6)

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

J2 = u0 u6 − 6u1 u5 + 15u2 u4 − 10u23 ,

u0 u1 u2 u3

u1 u2 u3 u4 

,

J4 = det 

5

 2

u3 u4 u5 u6

b0 b1 b2

J6 = det b1 b2 b3  ,

b2 b3 b4

J10 = u0 c − 6u1 bc2 + 3u2 (ac + 4b2 )c − 4u3 (3abc + 2b3 )

+ 3u4 a(ac + 4b2 ) − 6u5 a2 b + u6 a3 ,

where

b0 = 6(u0 u4 − 4u1 u3 + 3u22 ),

b1 = 3(u0 u5 − 3u1 u4 + 2u2 u3 ),

b2 = u0 u6 − 9u2 u4 + 8u23 ,

b3 = 3(u1 u6 − 3u2 u5 + 2u3 u4 ),

b4 = 6(u2 u6 − 4u3 u5 + 3u24 ),

a = 2(u0 u2 u6 − 3u0 u3 u5 + 2u0 u24 − u21 u6 + 3u1 u2 u5 − u1 u3 u4 − 3u22 u4

+ 2u2 u23 ),

b = u0 u3 u6 − u0 u4 u5 − u1 u2 u6 − 8u1 u3 u5 + 9u1 u24 + 9u22 u5 − 17u2 u3 u4

+ 8u33 ,

c = 2(u0 u4 u6 − u0 u25 − 3u1 u3 u6 + 3u1 u4 u5 + 2u22 u6 − u2 u3 u5 − 3u2 u24

+ 2u23 u4 ).

Appendix B

Weight Enumerators

We recall coding theory. A code C of length n means a subspace of Fn2 .

The weight wt(x) of x ∈ Fn2 means the number of nonzero xi . The inner

42

Nur Hamid, Masashi Kosuda and Manabu Oura

product of two elements x, y ∈ Fn2 is defined by

x · y :=

x i yi ∈ F 2 .

The dual code C ⊥ of C is defined by the subspace of Fn2 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

(g)

(g)

WC := WC (xa |a ∈ Fg2 ) =

n (c1 ,...,cg )

xa a

c1 ,...,cg ∈C a∈Fg2

where

na (c1 , . . . , cg ) = |{i | (c1i , . . . , cgi ) = a}|.

Let ρe be the combination of the Brou´e-Enguehard map T h and Igusa’s

homomorphism ρ. In other word, we can say

(g)

(g)

ρe(WC ) = ρ(T h(WC ))

for a code C. We omit the detail of ρe and only say that ρe maps the weight

enumerators in genus g to the ring S(2, 2g+2). The reader who is interested

in the detail of ρe can refer to [5]. For every code C used here, the expression

(g)

of ρe(WC ) is taken from [5].

We start with g = 1. The weight enumerators of some codes are related

to E-polynomials by the following relations.

ρe(We(1)

) = 2−1 ψ2 ,

ρe(Wg(1)

) = 2−5 · 11 · ψ23 − 2−3 · 7ψ6 .

24

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

Certain subrings in classical invariant theory

43

are the following.

ρe(We(2)

) = 2−4 ψ22 − 3 · 2−3 ψ4 ,

ρe(Wg(2)

) = 2−15 32 5−1 13−2 31−1 20129ψ26 − 2−14 513−2 31−1 59651ψ24 ψ4

24

+ 2−13 7 · 13−2 31−1 809ψ23 ψ6 − 2−9 3 · 5−1 7 · 11 · 31−1 ψ2 ψ10

+ 2−11 3 · 13−2 31−1 65287ψ22 ψ42 + 2−12 7 · 13−2 29 · 31−1 149ψ2 ψ4 ψ6

− 2−10 3 · 11ψ43 + 2−7 32 7 · 13−2 ψ62 ,

ρe(Wd+ ) = 2−10 5−1 13−2 31−1 6323ψ26 − 2−9 13−2 31−1 12143ψ24 ψ4

(2)

24

− 2−10 3 · 13−2 31−1 683ψ23 ψ6 + 2−6 3 · 5−1 11 · 31−1 ψ2 ψ10

+ 2−10 32 13−2 31−1 47 · 379ψ22 ψ42 − 2−9 13−2 31−1 2089ψ2 ψ4 ψ6

− 2−9 3 · 11ψ43 − 2−5 32 13−2 ψ62 ,

ρe(Wd+ ) = 2−16 5−1 13−2 31−1 20507ψ28 − 2−13 3−1 7 · 13−2 23 · 31−1 271ψ26 ψ4

(2)

32

− 2−11 5−1 13−2 23 · 31−1 227ψ25 ψ6 + 2−13 3−1 13−2 15541ψ24 ψ42

+ 2−9 3−1 13−2 31−1 4679ψ23 ψ4 ψ6 + 2−7 5−1 13−1 31−1 173ψ23 ψ10

− 2−11 7 · 13−2 31−1 27743ψ22 ψ43 − 2−6 13−2 31−1 139ψ22 ψ62

+ 2−9 3−1 13−2 31−1 2129ψ2 ψ42 ψ6 − 2−6 13−1 31−1 107ψ2 ψ4 ψ10

+ 2−12 3 · 43ψ44 + 2−5 13−2 31−1 281ψ4 ψ62

+ 2−1 3 · 5−1 13−1 31−1 ψ6 ψ10 ,

ρe(Wd+ ) = 31−2 (2−18 3−1 5−1 13−2 267941ψ210 − 2−16 3−1 13−2 606959 ψ28 ψ4

(2)

40

− 2−18 3−1 13−2 1877033 ψ27 ψ6 + 2−18 3−1 13−2 2812 541 ψ26 ψ42

+ 2−14 5−1 13−1 17 · 4871 ψ25 ψ10 + 2−17 3−1 13−2 2207 · 5779 ψ25 ψ4 ψ6

− 2−17 5 · 13−1 903827 ψ24 ψ43 − 2−14 3−1 5 · 13−2 107209 ψ24 ψ62

− 2−16 3−1 5 · 13−2 17 · 59 · 4957ψ23 ψ42 ψ6

− 2−12 7 · 13−1 7187 ψ23 ψ4 ψ10 + 2−16 3 · 5 · 13−2 37 · 205187ψ22 ψ44

+ 2−12 3−1 5 · 13−2 271919 ψ22 ψ4 ψ62 + 2−9 13−1 17 · 43ψ22 ψ6 ψ10

+ 2−15 5 · 7 · 13−2 79319ψ2 ψ43 ψ6 + 2−12 3 · 13−1 181 · 293 ψ2 ψ42 ψ10

− 2−15 33 19 · 312 ψ45 − 2−12 3−1 5 · 13−2 71 · 6719ψ42 ψ62

− 2−8 13−1 17 · 293ψ4 ψ6 ψ10 + 2−6 3 · 5−1 41ψ10

).

44

Nur Hamid, Masashi Kosuda and Manabu Oura

Nur Hamid

Mathematics Education,

Universitas Nurul Jadid, Probolinggo

INDONESIA

e-mail: hamidelfath@gmail.com

Masashi Kosuda

Yamanashi University

JAPAN

e-mail: mkosuda@gmail.com

Manabu Oura

Institute of Science and Engineering

Kanazawa University

JAPAN

e-mail: oura@se.kanazawa-u.ac.jp

(Received May 14, 2020)

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