Tgp chi khoa HQC TrUt/ng Dgi hgc Quy Nhan - Sd'3, T^p VII ndm 2013
O N T H E A C T I O N OF T H E S T E E N R O D - M I L N O R O P E R A T I O N S ON T H E I N V A R I A N T S OF T H E G E N E R A L LINEAR G R O U P S
NGUYEN THAI HOA", PHAM THI KIM MINH*, NGUYEN SUM=
1. INTRODUCTION
Let p be an odd prime number. Denote by GL„ = GL{n, Fp) the general linear group over the prime field Fp. Each subgroup of GL^ acts on the algebra P^ = £ ( 3 : 1 , . . . , x„) ®
^p{y\-^ • • • ^yn) in the usual manner. Here and in what follows, E{.,...,.) and F p ( . , . . . , . ) are the exterior and polynomial algebras over Fp generated by the indicated variables.
We grade Pn by assignmg dimXi = I and dimy, ^ 2.
Dickson showed in [1] that the invariant algebra Fp(yi,... ,yn)^^'^ is a polynomial algebra generated by the Dickson invariants Qn.s^ 0 < s < n. Huynh Miii [5, 6] com- puted the invariant algebras P^^". He proved that P^^" is generated by Qn,s, 0 < s <
(t, Pn,si,...,sfc, 0 < Sl ^ . . . < Sfe < n. Here Rn,si. ..s* are Mui invariants and Q„,s are Dickson invariants (see Section 2).
Let Ap be the mod p Steenrod algebra and let r^, ^j be the Milnor elements of dimen- sions 2p' — 1, 2p' — 2 respectively in the dual algebra At of Ap. In [4], Milnor showed that as an algebra,
A; = E{r^,r^,...) ® ¥p{iU2...-).
Then A' has a basis consisting of all monomials
with S = {si,...,Sfc), 0 < ^1 ^ . . . ^ Sk,R = {ru---,-^m), U > 0. Let St^-'^ £ Ap denote the dual of TS^ ''vith respect to that basis. Then Ap has a basis consisting of all operations St^'^. For 5 ^ 0 , / ? ^ (r), St^-'-''^ is nothing but the Steenrod operation P""
So, we call St^-^ the Steenrod-Milnor operation of type {S,R).
We have the Cartan formula
where J?i = (n,), R2 - (^2;), Ri + R2 ^ (n,-H r2,),5i D Sj - 0,u,^ € P„ (see Miii [6]) We denote 5t„ - 5t("'''°', 5 i ^ ' - St"'^-, where A, - ( 0 , . . . , 1 , . . . , 0) with 1 at the i-th place. In [6], Huynh Mui proved that as a coalgebra,
Ap = A{Sto, Sh,...) ® r(St^', St^\...).
6 NGUYEN THAI HOA", PHAM THI KIM MINH^ NGUYEN SUM"^
Here, A{Sto,Sti,...) (resp. ^ ( 5 i ' ^ ' , 5 / ' ^ ^ . . . ) ) denotes the exterior (reap, polynomial) Hopf algebra with divided powers generated by the primitive Steenrod-Milnor operations Sto, 5 t i , . . . (resp. St^\St^\...).
The Steenrod algebra Ap acts on P^ by means of the Cartan formula together with the relations
0, H = (0), 0,
Vk,
0,
S = (K), R = (0) otherivise.
5 = B, R = (0), 5 = 0, H = A „ otherwise, St^'^'x, •-
St'-'^y, -
for A; ^ 1, 2 , . . . ,n (see Steenrod-Epstein [8], Sum [11]). Since this action commutes with the action of GL^, it induces an actions of Ap on P^^".
The action of St^-^ on the modular invariants of subgroups of general linear group has partially been studied by many authors. This action for 5 ^ 0, P = (r) was explicitly determined by Smith-Switzer [7], Hung-Minh [2], Kechagias [3], Sum [11, 12, 13, 14).
Smith-Switzer [7], Wilkerson [15] have studied the action of 5 ( ^ ' on the Dickson invariants.
The purpose of the paper is to compute the action of the Steenrod-Milnor operations on the generators of p2^^. More precisely, we exphcitly determine the action of St^''^'' on the Dickson invariants (52,o and (32,i-
The analogous results for p = 2 have been anounced in [9].
The rest of the paper contains two sections. In Section 2, we recall some needed information on the invariant theory. In Section 3, we compute the action of the Steenrod- Milnor operations on the Dickson invariants.
2. P R E L I M I N A R I E S
Definition 2.1. Let (e^+i,... ,e„), 0 < A; < n, be a sequence of non-negative integers.
Following Dickson [1], Mui [5], we define
[k;ek+i,...,en] = -
ON THE ACTION OF THE STEENROD-MILNOR OPERATIONS ON THE INVARIANTS.... 7 in which there are exactly k rows of (xi... x„). (See the accurate meaning of the deter- minants in a commutative graded algebra in Mui [5]). For A; = 0, we write
[0;ei,,..,e„] - [ei,...,e„] =det{yf').
In particular, we set
in,s = [0,l,-- , s , - - - , ^ ] , 0 < s < n, L^^ L„,n = [ 0 , 1 , . . . , T l - 1].
Each [fc;e^+i,... ,e„] is an invariant of the special linear group SL^ and [ei,...,e„] is divisible by !.„. Then, Dickson invariants Qn,s^ Q < s < n, are defined by
Here, by convention, LQ = [0] — 1.
Now we recall the following which will be used in the next section.
Lemma 2.2 (Sum [12]). The actions of St^ on L2.-^2.0.-^2,1 for £{R) — 2 are respectively given in the following tables:
(hi) (0,0) (O.P) (l,p) ( p + 1 . 0 )
5f("lL2 L2
^2{Q2,l ~ Q2.0) L2Q2.DQ2.I L2Q2.O
(i,3) (0,1) ( 0 , P + 1 )
(P,0) otherwise
St^^'^^L^
—L2Q2.0
•^2<52,0 L2Q2,l 0,
(',]) S&-'^L2.o {',]) 5i'''JlL2,o
(0,0) (0,P') (p.p') p^+p,0)
L2Q2.0 L2(Q2.oQ2.1
^2Q2.0 Q2.I L2Q2Q
(0,P) (O.p'+P)
ip',0) otherwise
" ^ ' 2 ^ 2 , 0 -t'2V2,0 L2Q2,oQ2 0,
NGUYEN THAI HOA", PHAM THI KIM MINH', NGUYEN SUM'
(i.j)
(0,0) (0,p') [O.p' + 1)
(1,0) ilp') (p\o) (PM) (p' + l,0)
otherwise
5i<''^'L2,i L2Q2.1
^iiQl,! ^ "" '52,0*52,1
^20/2.0 V2,i
^ 2 ^ 2 , 0
i'2(Q2,oQ2,l ^ ~ ^2,0 )
i2(Qs:;'-Q;,„)
^2*52.0
•^202,oQ2,l
0.
— Q 2 , 0 ^ 2 , l )
3. M A I N R E S U L T S First of all, we recall the following.
P r o p o s i t i o n 3.1 (Hurng-Minh [2], Sum [11]). Let i be a nonnegative integer. We have
[0?,o. i = p ' - l ,
St"'°'Q2,o = < ( - l ) ' ( r ) Q S ' Q 2 , 7 , i = /tp + r, 0 < r < ;t < p,
\0, otherwise,
foil. i = p'-P,
5t'«'Q2,i = I ( - l ) ' ( ' ; ; - ' ) Q 5 , „ Q 5 t ' " ' . i = *P + r, r < A: + 1 < p, [ 0 , otherwise.
Theorem 3.2 (Sum [12]). Let j be a nonnegative integer and s ^ 0 , 1 . Then we have
iQi.oQ2.. E'=„(-i)' ( S ) Cf)Q2'o-'''Q?,r",
St*°'^'Q2,, if j = kp -\-r with 0 < fc, r < p, otherwise.
Our new results are the actions of 5t'''^' on Q2,o and Q2,i with i > 0.
Theorem 3.3. Let i,j be nonnegative integers. We have (i) / / 0 < fc < i < p anrf 0 < r < p then
(ii) IfO<i<p then
5f<'''P+^'Q2,o - 0.
St^''''^Q2.o'^ {-lYQifQti-
ON THE ACTION OF THE STEENROD-MILNOR OPERATIONS ON THE INVARIANTS.... 9
-Q;?Q?,iE':„'(-i)'[M'7')r:;!') -K:;)(;:;)]o^'o"'"'"Q?,r".
if j = kp + r with 0 < fc, r < p, 0, otherwise.
(iv) / / fc -I- 1 < 2 < p and 0 < r < p then
(v) 7/0 < fc < p i/iera (v.)
[ (fc+1)^2? Eio(-i)' 0 (T)'55r'''02*r''
5t*^-^'Q2,i = < ifj^kp + r with 0 < fc, r < p, I 0, otherwise.
Proof We prove i) by induction on fc. Let k ^ 0. For r = 0, using Cartan formula we have Sv^'°'Q2fi = 0. For r > 0, applying the Cartan formula and the inductive hypothesis one gets
5t<'-^'(52,o - Q2.o5't''''"-^'Q2,o = 0.
Part i) of the theorem holds for fc ^ 0. Suppose it is true for 0 < fc < p — 1. For i > fc -I-1, we have
5f"*+"'''Q2,„ = Q2.oSt''*'+'-'>Q2,c + (Q|,„ - Q?J')5f'-'''''Q2,„
- Q;J'St(-.<'-'>''+'-'>Q2,„ - Q 2 , O Q ; , , 5 ( ( - ' - " > Q 2 , „
= 0.
Part i) of the theorem holds for i > fc + 1 and r = 0. Suppose it is true for i > fc + 1 and 0 < r < p — 1. Then using the inductive hypothesis we have
s(<''"+"''+'+"«2,o = Q2,oSt<'*+"''+''Q2,„ + (Q;,„ - Q;:i')5('-.'"'+'+'>02,o
- Q;J'St<''"+'>Q2,o - Q2,oQ;,,St<'-''''+'+"Q2,„
= 0.
Part i) of the theorem is proved.
10 NGUYEN THAI HOA°, PHAM THI KIM MINH', NGUYEN SUM=
Now we prove Part ii). Obviously, it is true for i = 0. Suppose i > 0 and Part ii) holds for z - 1 Using the Cartan formula and the inductive hypothesis, we have
5i<''"'02,o = -Q2,oQ;,i5("-''"-"'''g2.o
= -(-i)-'Q2,oO;,iQ-2,„(2&"''
= {-iYQ'2VQt-
The proof of Part iii). It is well known that the Steenrod-Milnor operations are stable. That means St^^'^^x = 0 for any homogeneous x e P2 with degz < j (see Miii [5]).
Since degQa.o ^ p^ - i < p^, we have 5('''-''Q2,D = 0 for j > p^. Suppose that j < p^.
Then using the p-adic expansion of j , we have j = kp -\- r for 0 < fc, r < p.
Now we prove the theorem by double induction on (fc,r).
If fc = 0, then using Part i) of Theorem 3.3 we get 5f''''"'Q2,o = 0, so the theorem is true for fc - 0. If r = p - 1, then Cl^j^) -- {[^'^ - 0 in Fp, so we obtain
If r — 0, then the formula in the theorem becomes
St^''''^Q2,o = ~kQl^Ql,{Ql, - (T^!,')'-'
We prove this formula by induction on fc. Suppose this formula holds for 0 < fc < p - 1.
For r = 0, we have
St<'*+i)rtQ, „ = ( Q ; „ _ Q;+')SiC'")02.„ - Q2.o(??,,S(<°-"''e2,o
= —fcQ2,o'52.l{'32.0 ~ Q2,l ) ^ Q2.0*52,l(*52,0 ~ *52,1 )
= -(k + l)Ql„Ql,{Ql,-Q',y)':
So the formula holds for fc -I-1 and r = 0. Suppose 0 < r < p — 1 and the formula holds for j = {k+ l)p + r. Applying the Cartan formula and the inductive hypothesis, one gets
5t'«'+"''+'+"Q2.o = Q2,o5t"*+'>+'-'(32,o+(m.o - Q;:;')5t<'''"-+'-+"Q2,o - Qto'S(<'''"'+''Q2,o - Q2,oQ;,iSf(°.'-+'+"Q2,„
ON THE ACTION OF THE STEENROD-MILNOR OPERATIONS ON THE INVARIANTS.... 11
=-«s=oii(^D-)-[0(-:r)-fr)f::r)
-£-)'[C:0(:::)^C:Oft:tO
-C")t:')-C:i)(::i)]«»'"«^'")
= -Q5?Q?,iE(-')'[(* + i ) ( - ) ( ' t i t ' )
'°-C:i)f::r)i«a-"«^r.
Parts iv), v) and vi) are proved by the same argument as the previous one.
D
This work was supported in part by a grant of Quy Nhon University Research Project.
R E F E R E N C E S
[1] L. E. Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), 75-98.
[2] N.H.V. Hung and P.A. Minh, The action of the mod p Steenrod operations on the modular invariants of linear groups, Vietnam J. Math. 23 (1995), 39-56.
[3] N E Kechagias, The Steenrod algebra action on generators of rings of invariants of subgroups o/GL„(Z/pZ), Proc. Amer. Math. Soc. 118 (1993), 943-952.
[4] J. Milnor, Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150-171.
[5] H. Miii, Modular invariant theory and the cohomology algebras of symmetric groups, 3. Fac. Sci. Univ. Tokyo Sec. IA Math 22 (1975), 319-369.
[6] H. Miii, Cohomology operations derived from modular invariants, Math. Z. 193 (1986), 151-163.
[7] L, Smith and R. Switzer, Reahzability and non-realizability of Dickson algebras as cohomology rings, Proc. Amer. Math. Soc. 89 (1983), 303-313.
[8] N.E. Steenrod and D.B.A. Epstein, Cohomology operations, Ann. of Math No. 50, Princeton University Press, 1962.
12 NGUYEN THAI HOA°, PHAM THI KIM MINH*, NGUYEN SUM=
[9] N. Sum, On the action of the Steenrod-Milnor operations on the modular invariants of linear groups, Japan. J. Math. 18 (1992), 115-137.
[10] N. Sum, On the action of the Steenrod algebra on the modular invariants of special linear group, Acta Math. Vietnam. 18 (1993), 203-213.
[11] N Sum, Steenrod operations on the modular invariants, Kodai Math. J. 17 (1994), 585-595.
[12] N Sum, On the action of the Steenrod-Milnor operations on the Dickson invariants, Vinh University Journal of Science 33 (2004), 41-48.
[13] N. Sum, On the module structure over the Steenrod algebra of the Dickson algebra, Quynhon University Journal of Science i (2007), 5-15.
[14] N. Sum, The action of the primitive Steenrod-Milnor operations on the modular in- variants, Geom. Topol. Monogr., Math. Sci. Publ., Coventry, l l (2007), 349-367.
[15] C. Wilkerson, A primer on the Dickson invariants, Contemporary Mathematics, Amer. Math. Soc. 19 (1983), 421-434.
TOM TAT
VE TAC DONG CUA CAC TOAN TLl STEENROD-MILNOR TREN CAC BAT BIEN CUA NHOM TUYEN TIXH TONG QUAT
Nguyen Thai Hoa, Pham Thi Kim Minh, Nguyen Sum Cho p la mot so nguyen to 1^ va Fp la truang nguyen to co p phan tii Trong bai bao nay, chiing toi ti'nh tac dong ciia cac toan tii Steenrod-Milnor do dai 2 tren cac phan tli sinh cua dai so cac bat bien ciia nhdm tuyen tfnh tong quat GI-(2,Fp) trong dai so da thiic hai bien.
"'•^Khoa Toan, Trucrng Dai hoc Quy Nhcfn
^Hoc vien Cao hpc K. 12, Khoa Toan, Truo-ng Dai hoc Quy Nho-n Ngay nhan bai: 03/6/2013; Ngay nhan dang: 18/7/2013.