> # This program computes the Krieg matrix. # IF SM is the spherical image of the T_*(p^2) Hecke operators # and X is the W_n-invariant polys [x^b] then # SM= X K_n(p^2). # So K_n(p^2)) is an (n+1)x(n+1) matrix. # # The row vector SM is actually listed T_n(p^2),...,T_0(p^2). # and the row vector X is listed [x11...1,x21...1,x22...1,...,x22...2]. # # The procedure Kmatrix(n) below will output p^( n(n+1) )*K_n(p^2). # The entries of Kmatrix(n) are all polynomials in p. # Divide by p^( n(n+1) ) to get the entries of K_n(p^2). # # This program was written for "Lifting Puzzles in Degree 4" # by C. Poor, N. Ryan and D. Yuen. # Compare Theorem 3.1. # # C. Poor wrote this program. # The prime p is a global variable. Sorry. # # ## EXAMPLE: >Kmatrix(4); > krieggamP:=proc(L,k) local S,F,G,t; global p; S:=1; F:=1;G:=1; if L=0 then S:=1; fi; if L=1 then S:=(p^(2*k+2)-1)/(p^2-1) +p^k; fi; if L>1 then for t from 1 to L do F:=F*( (p^(2*k+2*t)-1 )/( p^(2*t) -1) ) od: for t from 1 to (L-1) do G:=G*( (p^(2*k+2*t)-1 )/( p^(2*t) -1) ) od: S:=F+(p^k)*G; fi; simplify(S); end: kriegCeven:=proc(L,k) local C,S,F,G,t; global p; C:=1;S:=0; if (L mod 2)= 0 then for t from 0 to (L/2) do S:=S+(-1)^t*binomial(L+k,L/2-t)*krieggamP(t,k)*p^(L^2+2*L*k+L+k*(k+1)/2-2*k*t-t^2) od: fi; simplify(S); end: kriegC:=proc(L,k) local C,S,F,G,t; global p; C:=1;S:=0; if (L mod 2)= 0 then for t from 0 to (L/2) do S:=S+(-1)^t*binomial(L+k,L/2-t)*krieggamP(t,k)*p^(L^2+2*L*k+L+k*(k+1)/2-2*k*t-t^2) od: fi; if (L mod 2)= 1 then S:=(p^(k+1)-1)*kriegCeven(L-1,k+1) fi; simplify(S); end: krieg:=proc(a,j,k) local C,S,F,G,t; global p; S:=1; S:= kriegC(j-a,k)*p^(a*(2*(j+k)+1-a)); S; end: kriegmatrix:=proc(n) local f,i,j; global p; f:= (i,j) -> krieg(i-1,j-1,n+1-j): Matrix(n+1,f); end: Kmatrix:=proc(n) local f,i,j; global p; if n=1 then lprint(`The Kreig matrix isn't appropriate for n=1.`) fi; kriegmatrix(n); end: > Kmatrix(4); ## This is the Krieg matrix p^(20)*K_4(p^2) for degree n=4. [[ 10 / 4 \ 10 17 16 14 12 [[p , \p - 1/ p , 4 p - p - 2 p - p , / 2 \ 12 / 5 4 2 \ 20 19 16] [ 14 \p - 1/ p \4 p - p - 2 p - 1/, 6 p - 8 p + 2 p ], [0, p , / 3 \ 14 / 11 10 9 8\ 8 16 / 3 2 \] \p - 1/ p , \3 p - p - p - p / p , (p - 1) p \3 p - p - p - 1/], [ 17 / 2 \ 17 / 6 5\ 14] [ 19 19] [0, 0, p , \p - 1/ p , \2 p - 2 p / p ], [0, 0, 0, p , (p - 1) p ], [ 20]] [0, 0, 0, 0, p ]] > Kmatrix(4)[2,4]; ## This is the (2,4)-entry of the above matrix Kmatrix(4). / 11 10 9 8\ 8 \3 p - p - p - p / p