# # qmult - q-analogue of weight multiplicities in representations # # Calling sequence: # qmult(v,R); # qmult(v,u,R); # # Parameters: # R = a crystallographic root system data structure # u,v = dominant integral weights (linear combinations of e1,e2,...) # # There is a polynomial in q associated to each pair of dominant integral # weights u and v with u a weight of the irreducible representation of # LieAlg(R) of highest weight v (i.e., u a member of weight_sys(v,R)). # It was defined originally by Lusztig as a q-analogue of Kostant's weight # multiplicity formula, and in particular it gives the dimension of the # u-weight space of the representation (i.e., weight_mults(v,u,R)) at q=1. # Also, with coefficients reversed, it is a Kazhdan-Lusztig polynomial for # the associated affine Weyl group. # # Reference: R. K. Gupta, J. London Math. Soc. 36 (1987), 68--76. # # If u and v are dominant weights as above, qmult(v,u,R) returns the # corresponding q-analogue of weight multiplicity. With two arguments, # qmult(v,R) computes the q-analogue for each dominant weight u in # weight_sys(v,R). The output is expressed as a linear combination of the # form f1*M[w1] + f2*M[w2] + ... , where w1,w2,... are the weight # coordinates of the dominant weights that appear, and f1,f2,... are the # q-analogues of their weight multiplicities. # # The algorithm is a slight modification of Broer's algorithm (Indag. Math. # 6 (1995), 385--396), and proceeds by solving a triangular system of # equations in a space of dimension (1+a1)*(1+a2)*..., where [a1,a2,...] # are the simple root coordinates of v - u0, with u0 = u (if u is given), # or u0 = the lowest dominant weight in weight_sys(v,R) (otherwise). In # particular, the space requirements of the procedure are much greater # than those of the 'weight_mults' function in the weyl package. # # Warning: both M and q are used as global names by this procedure. # # Examples: # with(coxeter): with(weyl): # qmult(rho(B3),B3); # w:=weights(F4): qmult(w[4],0,F4); # qmult:=proc(u) local S,wt,wtsys,wts,wc,r0,mu,nu,K,pr,sat,f,r,beta,k,w,J; S:=coxeter['base'](args[nargs]); wt:=weyl['weights'](S); pr:=coxeter['pos_roots'](S); r0:=convert(pr,`+`)/2; if nargs>2 then mu:=args[2]; r:=coxeter['root_coords'](u-mu,S); if not type(r,'list'('nonnegint')) then RETURN(0) fi else wtsys:=weyl['weight_sys'](u,S,'wc'); mu:=wtsys[nops(wc)] fi; wts:=[u-mu]; K[wts[1]]:=1; for sat while sat<=nops(wts) do J:=map(proc(x,Y,Z,y) if coxeter['iprod'](Y[x],y)>0 then y-Z[x] fi end, [\$1..nops(S)],wt,S,wts[sat]); for nu in J do if member(nu,wts) then next fi; wts:=[op(wts),nu]; f:=0; for r in pr do beta:=nu+r; for k from 0 while member(beta,wts) do f:=f+q^k*K[beta]; beta:=beta+r od od; K[nu]:=int(f,q); if coxeter['vec2fc'](nu+mu+r0,S,'w')=u+r0 then K[nu]:=K[nu]+(-1)^nops(w) fi; od od; if nargs>2 then K[0] else convert([seq(K[wtsys[k]-mu]*M[op(wc[k])],k=1..nops(wc))],`+`) fi end;