Home /
Expert Answers /
Advanced Math /
consider-an-to-be-the-subgroup-of-upper-triangular-matrices-whose-diagonal-entries-can-be-any-n-pa373
(Solved): Consider An (to be the subgroup of Upper triangular matrices whose diagonal entries can be any n ...
Consider An (to be the subgroup of Upper triangular matrices whose diagonal entries can be any non zero elements from Fp.) in Gln(Fp) (n by n matrices with non zero determinant and (Fp is finite field with p elements)). Let Vn be the vector space of C - valued function on Gln(Fp) (i.e these are functions from Gln(Fp)→C ) that are invariant with respect to left and right An-translations. We introduce a binary operation on Vn as follows. Given two bi invariant functions, define (f1⋆f2)(g)=∣An∣1∑x,y∈GLn,xy=gf1(x)f2(y). Consider V3 with the convolution product. Denote the function which vanishes outside A3⊂GL3(Fp) and equals 1 otherwise by Le. Let P1 (resp. P2 ) be subgroup in GL3(Fp) given by a31=a32=0( resp,a21=a31=0). Denote the function which vanishes outside P1⊂GL3 and equals 1 otherwise by L1′, the characteristic function of P2 is L2′. Put L1=L1′−Le and L2=L2′−Le. We are given that the following relations hold. L1⋆L1=(p−1)L1+pLeL2⋆L2=(p−1)L2+pLeL2⋆L1⋆L2=L1⋆L2⋆L1 Prove that the elements T1,T2 generate V3