A note on the action of the primitive Steenrod-Milnor operations on the Dickson invariants
aa r X i v : . [ m a t h . A T ] J a n A NOTE ON THE ACTION OF THE PRIMITIVESTEENROD-MILNOR OPERATIONS ONTHE DICKSON INVARIANTS
NGUYỄN SUM
Abstract.
In this note, we present a formula for the action of the primitiveSteenrod-Milnor operations on generators of algebra of invariants of the generallinear group GL n = GL ( n, F p ) in the polynomial algebra with p an arbitraryprime number. Introduction
Let p be a prime number. Denote by GL n = GL ( n, F p ) the general lineargroup over the prime field F p of p elements. This group acts on the polynomial P n = F p [ x , x , . . . , x n ] in the usual manner. We grade P n by assigning dim x j = 1for p = 2 and dim x j = 2 for p >
2. Dickson showed in [1] that the invariantalgebra P GL n n is a polynomial algebra generated by the Dickson invariants Q n,s with 0 s < n .Let A ( p ) be the mod p Steenrod algebra and denote by St R ∈ A ( p ) the Steenrod-Milnor operation of type R , where R is a finite sequence of non-negative integers(see Milnor [2], Mùi [3, 4]). For R = ( k ), St ( k ) is the Steenrod operation P k .For ∆ i = (0 , . . . , ,
1) of length i , St ∆ i is called the i -th primitive Steenrod-Milnoroperation in A ( p ).The Steenrod algebra A ( p ) acts on P n by means of the Cartan formula togetherwith the relation P k ( x j ) = ( x pj , if k = 1 , , otherwise,for j = 1 , , . . . , n (see Steenrod-Epstein [6]). Note that for p = 2, P k is theSteenrod square Sq k . Since this action commutes with the one of GL n , it inducesan inherited action of A ( p ) on P GL n n .The action of St ∆ i on the modular invariants of linear groups has partially beenstudied by Smith-Switzer [5], Sum [8] and Wilkerson [9]. The purpose of the note isto present a new formula for the action of the primitive Steenrod-Milnor operationson the Dickson invariants. Mathematics Subject Classification.
Primary 55S10; Secondary 55S05.
Key words and phrases.
Polynomial algebra, Steenrod-Milnor operations, modular invariants. Main Result
First of all, we introduce some notations. Let ( e , . . . , e n ) be a sequence ofnon-negative integers. Following Dickson [1], we define[ e , e , . . . , e n ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x p e · · · x p e n x p e · · · x p e n ... · · · ... x p en · · · x p en n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Denote L n,s = [0 , , . . . , ˆ s, . . . , n ] , s n, L n = L n,n = [0 , , . . . , n − . Each[ e , e , . . . , e n ] is divisible by L n and [ e , e , . . . , e n ] /L n is an invariant of GL n . ThenDickson invariants Q n,s are defined by Q n,s = L n,s /L n , s < n. By convention, Q n,s = 0 for s <
0. Note that Q n, = L p − n . Theorem 2.1 (See Dickson [1]) . P GL n n = F p [ Q n, , Q n, , . . . , Q n,n − ] . The main result of the note is the following.
Theorem 2.2.
For any s < n and i > , we have St ∆ i ( Q n,s ) = ( − n Q n, ( P pn,i,s + R pn,i Q n,s ) , where P n,i, = 0 , P n,i,s = [0 , . . . , [ s − , . . . , n − , i − /L n , for s > and R n,i = [0 , , . . . , n − , i − /L n . The motivation of expressing St ∆ i ( Q n,s ) in this theorem in a different way fromits original description in Theorem 2.5 derives from a result in [8], which was usedby N. H. V. Hưng from the Vietnam Institute for Advanced Study in Mathematics,in his talk in October 2019, on “The Margolis homology of the Dickson algebra”.We need the following results for the proof of the theorem. Theorem 2.3 (See Smith-Switzer [5], Wilkerson [9]) . For any s < n and i n , we have St ∆ i ( Q n,s ) = ( − s − Q n, , i = s > , ( − n Q n, Q n,s , i = n, , otherwise. Theorem 2.4 (See Sum [7, 8]) . For any sequence ( e , . . . , e n ) of non-negativeintegers, we have [ e , . . . , e n − , e n + n ] = n − X s =0 ( − n + s − [ e , . . . , e n − , e n + s ] Q p en n,s . Theorem 2.5 (See Sum [8]) . For any s < n and i > , we have St ∆ i ( Q n,s ) = ( − n [0 , , . . . , ˆ s, . . . , n − , i ] L p − n . We now prove Theorem 2.2.
HE ACTION OF THE PRIMITIVE STEENROD-MILNOR OPERATIONS 3
Proof of Theorem 2.2.
By Theorem 2.5, we have St ∆ i ( Q n, ) = ( − n [1 , , . . . , n − , i ] L p − n = ( − n (cid:0) [0 , , . . . , n − , i − /L n (cid:1) p L p − n = ( − n R pn,i Q n, . Hence, the theorem is true for s = 0.Assume that s >
0. We prove the theorem by induction on i . By Theorem 2.3,the theorem is true for 1 i n . Suppose that i > n and the theorem holds for1 , , . . . , i . Using Theorems 2.4, 2.5 and the inductive hypothesis, we get St ∆ i +1 ( Q n,s ) = ( − n [0 , , . . . , ˆ s, . . . , n − , i + 1] L p − n = n − X t =0 ( − t − [0 , . . . , ˆ s, . . . , n − , i − n + 1 + t ] Q p i − n +1 n,t L p − n = n − X t =0 ( − n + t − St ∆ i − n +1+ t ( Q n,s ) Q p i − n +1 n,t = n − X t =0 ( − t − Q n, ( P pn,i − n +1+ t,s + R pn,i − n +1+ t Q n,s ) Q p i − n +1 n,t = ( − n Q n, (cid:16) n − X t =0 ( − n + t − P pn,i − n +1+ t,s Q p i − n +1 n,t + (cid:16) n − X t =0 ( − n + t − R pn,i − n +1+ t Q p i − n +1 n,t (cid:17) Q n,s (cid:17) . Using Theorem 2.4, we have n − X t =0 ( − n + t − P pn,i − n +1+ t,s Q p i − n +1 n,t = n − X t =0 ( − n + t − (cid:0) [0 , . . . , [ s − , . . . , n − , i − n + t ] /L n (cid:1) p Q p i − n +1 n,t = (cid:16)(cid:16) n − X t =0 ( − n + t − [0 , . . . , [ s − , . . . , n − , i − n + t ] Q p i − n n,t (cid:17) /L n (cid:17) p = (cid:0) [0 , . . . , [ s − , . . . , n − , i ] /L n (cid:1) p = P pn,i +1 ,s . By a similar computation using Theorem 2.4, we obtain n − X t =0 ( − n + t − R pn,i − n +1+ t Q p i − n +1 n,t = n − X t =0 ( − n + t − (cid:0) [0 , , . . . , n − , i − n + t ] /L n (cid:1) p Q p i − n +1 n,t = (cid:16)(cid:16) n − X t =0 ( − n + t − [0 , , . . . , n − , i − n + t ] Q p i − n n,t (cid:17) /L n (cid:17) p = (cid:0) [0 , , . . . , n − , i ] /L n (cid:1) p = R pn,i +1 . NGUYỄN SUM
Thus, the theorem is true for i + 1. So, the proof is completed. (cid:3) Using Theorems 2.2 and 2.4, we can explicitly compute the action of St ∆ i onthe Dickson invariants Q n,s for i > n by explicitly computing P n,i,s and R n,i . Thecases i = n + 1 , n + 2 have been computed in Sum [8] by using Theorem 2.5. Corollary 2.6 (See Sum [8]) . For s < n , we have St ∆ n +1 ( Q n,s ) = ( − n Q n, ( − Q pn,s − + Q pn,n − Q n,s ) ,St ∆ n +2 ( Q n,s ) = ( − n Q n, (cid:0) Q p n,s − − Q pn,s − Q p n,n − + ( Q p + pn,n − − Q p n,n − ) Q n,s (cid:1) . By a direct calculation using Theorem 2.4, we easily obtain the following.
Corollary 2.7.
For s < n , St ∆ n +3 ( Q n,s ) = ( − n Q n, ( P pn,n +3 ,s + R pn,n +3 Q n,s ) ,where P n,n +3 ,s = Q p n,s − − Q pn,s − Q p n,n − − Q n,s − Q p n,n − + Q n,s − Q p + pn,n − ,R n,n +3 = Q p n,n − − Q p n,n − Q n,n − − Q pn,n − Q p n,n − + Q p + p +1 n,n − . Corollary 2.8.
For any s < n and i > , we have St ∆ i ( Q p − n, Q n,s ) = ( − n Q pn, P pn,i,s ∈ Ker( St ∆ i ) . Proof.
Using the Cartan formula for the Steenrod-Milnor operations (see Mùi [3, 4])and Theorem 2.2, we get St ∆ i ( Q p − n, Q n,s ) = ( p − Q p − n, St ∆ i ( Q n, ) Q n,s + Q p − n, St ∆ i ( Q n,s )= ( − n (cid:0) − Q pn, R pn,i Q n,s + Q pn, ( P pn,i,s + R pn,i Q n,s ) (cid:1) = ( − n Q pn, P pn,i,s . The corollary is proved. (cid:3)
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Department of Mathematics and Applications, Sài Gòn University, 273 An DươngVương, District 5, Hồ Chí Minh city, Viet Nam
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