# # LR_rule - implementation of the Littlewood-Richardson rule # # Calling Sequence: # LR_rule(lambda,mu); # LR_rule(lambda,mu,alpha); # LR_rule(lambda,mu,alpha,beta); # # Parameters: # lambda,mu,alpha,beta - partitions # # With two arguments, LR_rule(lambda,mu) computes the expansion of the skew # Schur function indexed by lambda/mu via the Littlewood-Richardson rule. # This is not necessarily faster than using the built-in commands tos and # jt_matrix in the SF package, but is provided for the sake of comparison. # # With a third argument alpha, it computes the coefficient of s[alpha] in # the skew Schur function indexed by lambda/mu. This is also the scalar # product of s[lambda] and s[mu]*s[alpha]. # # With a fourth argument beta, it computes the scalar product of the skew # Schur functions indexed by lambda/mu and alpha/beta. # # Reference: # I. Macdonald, "Symmetric Functions and Hall polynomials", Section I.9. # # Examples: # LR_rule([6,5,4,3,2,1],[4,3,3,1]); # LR_rule([6,5,4,3,2,1],[4,3,3,1],[4,3,1,1,1]); # LR_rule([8,6,3,2],[6,3,2],[6,6,4,3,2],[5,3,3,2]); LR_rule:=proc(lambda) local l,mu,alpha,beta,i,j,dgrm; if not `LR_rule/fit`(lambda,args[2]) then RETURN(0) fi; l:=nops(lambda); mu:=[op(args[2]),0\$l]; dgrm:=[seq(seq([i,-j],j=-lambda[i]..-1-mu[i]),i=1..l)]; if nargs>2 then alpha:=args[3]; if nargs>3 then beta:=args[4] else beta:=[] fi; if not `LR_rule/fit`(alpha,beta) then RETURN(0) fi; l:=convert([op(lambda),op(beta)],`+`); if l<>convert([op(alpha),op(mu)],`+`) then RETURN(0) fi; nops(LR_fillings(dgrm,[alpha,beta])) else convert([seq(s[op(i[1])],i=LR_fillings(dgrm))],`+`) fi end; # Generate all LR-fillings of the given diagram. # The output is a list of pairs [nu,lp], where lp is a lattice permutation, # and nu is its "shape". If there is a second argument [alpha,beta], then # the output is the list of such pairs that are compatible with the skew # shape alpha/beta. In the case beta=[], "compatible" means that nu=alpha. LR_fillings:=proc(dgrm) local n,x,upper,lower; if dgrm=[] then if nargs=1 then x:=[] else x:=args[2][2] fi; RETURN([[x,[]]]) fi; n:=nops(dgrm); x:=dgrm[n]; if not member([x[1],x[2]+1],dgrm,'upper') then upper:=0 fi; if not member([x[1]-1,x[2]],dgrm,'lower') then lower:=0 fi; if nargs=1 then map(`LR/nextletter`,LR_fillings([op(1..n-1,dgrm)]),lower,upper) else map(`LR/nextletter`,LR_fillings([op(1..n-1,dgrm)],args[2]), lower,upper,args[2][1]) fi; end: `LR/nextletter`:=proc(T) local shape,lp,lb,ub,i,nl; shape:=[op(T[1]),0]; lp:=T[2]; ub:=nops(shape); if nargs>3 then ub:=min(ub,nops(args[4])) fi; if args[2]=0 then lb:=0 else lb:=lp[args[2]] fi; if args[3]>0 then ub:=min(lp[args[3]],ub) fi; if nargs<4 then nl:=map(proc(x,y) if x=1 or y[x-1]>y[x] then x fi end,[\$lb+1..ub],shape) else nl:=map(proc(x,y,z) if y[x]y[x]) then x fi end, [\$lb+1..ub],shape,args[4]) fi; nl:=[seq([subsop(i=shape[i]+1,shape),[op(lp),i]],i=nl)]; op(subs(0=NULL,nl)) end: `LR_rule/fit`:=proc(lambda,mu) local i,l; l:=nops(mu); if l>nops(lambda) then RETURN(false) fi; for i to l do if mu[i]>lambda[i] then RETURN(false) fi od; true end: