The Connection Between The Number Of Subsystems With Rank L In An Irreducible Root System And The Chevalley Groups

In this paper,we investigate some irreducible root systems which are determinedby the simple lie algebra,the number of the irreducible subsystems with rank l iscomputed,and have the following results:Theorem 2.1 Let be an irreducible root system of type D_n(n≥4),then thenumber of the irreducible subsystems with rank l of type of A_n isTheorem 2.2 Let be an irreducible root system of type D_n(n≥4),then thenumber of the irreducible subsystems with rank l of type of D_l(l≥4)is g′_l(D_n)=C_n^l.Theorem 2.3 Let be an irreducible root system of type D_n(n≥4),then there are only irreducible root systems of type A_n,D_n in,and the number of theirreducible subsystems with rank l of isTheorem 3.1 Let be an irreducible root system of type B_n(n≥4),then thenumber of the irreducible subsystems with rank l of type of B_l is g″_l(B_n)=C_n^l. Theorem 3.2 Let be an irreducible root system of type B_n(n≥4),then thenumber of the irreducible subsystems with rank l of isTheorem 3.3 Let be an irreducible root system,and ^v be the dual root system of ,then there is a bijection between the irreducible subsystems with rank l of and the irreducible subsystems with rank l of ^v,specially,G_l(B_n)=G_l(C_n).Theorem 3.4 Letε₁,ε₂,...,ε_n+1 be standard orthogonal basis of Euclidean space Rⁿ⁺¹,any l+1 vectorsε_i1,ε_i2,...,ε_il+1(1≤l≤n)of Euclidean space Rⁿ⁺¹,and theconsists irreducible root subsystem of type A_l,specially,G_l(A_n)=C_n+1^l+1.Theorem 4.1 Let be an irreducible root system of typeΠ,Π(K)be Chevalley group of typeΠon field K,andΦ_l⁽ⁱ⁾ be an irreducible root subsystem of with rank l.then for any irreducible root subsystemΦ_l⁽ⁱ⁾,there is subgroupΠ_l⁽ⁱ⁾(K)inΠ(K),andΠ_l⁽ⁱ⁾ be the type ofΦ_l⁽ⁱ⁾,i=1,2,...,G_l(Π).