> 2+2 4 > 2+2; 3+3 6 > 2+2; \ 4 > ?orbit orbit(int,vec,mat)-> mat orbit(n,v,M) [result: vectors]. This function operates in the same way as orbit(v,m), but n replaces the limit of 1000 elements in the orbit. Warning: orbit uses allocates space at the beginning for the maximal number n of vectors allowed in the orbit; therefore one shouldn't go overboard on choosing the limit n. orbit(vec,mat)-> mat orbit(v,M) [result: vectors]. Here v is a vector with an arbitrary interpretation, and M is a matrix whose column size c equals size(v), and whose row size is a multiple of c, say kc. We interpret M as a collection of k square matrices of size c x c, vertically concatenated. The function orbit attempts to compute the orbit of v under the group generated by the collection of matrices, i.e., a minimal set V of vectors containing v and closed under right multiplication by any of the matrices in the given collection. As the orbit might be infinite, and the algorithm has no means to detect this situation, it gives up when more than 1000 vectors in the orbit have been computed. For larger orbits, see orbit(n,v,M ), for Weyl group orbits see W_orbit. > ?KL There is no information about the term 'KL'. > ?KLpoly There is no information about the term 'KLpoly'. > ?KLpoly KL_poly(vec,vec,grp)-> pol KL_poly(x,y,g) [x,y: Weyl word, result: polynomial]. Returns the Kazhdan-Lusztig polynomial P_{x,y}. > W_word([1,2,3,3],A3) [1,2] > reduce([1,3,2,3,2],A3) [1,2,3] > setdefault D4 > reduce([1,2,3,4,3,4,2,1,4]) [4] > reduce([1,2,3,4,3]) [1,2,4] > long_word [1,2,1,3,2,1,4,2,1,3,2,4] > KL_poly([1,2,1],long_word) 1X[0] > KL_poly([3,2],[1,2,1,3,2,1,4,2,1,3,2]) 1X[0] +1X[2] > center [[0,0,1,1,2] ,[1,0,1,0,2] ] > print_tab(1,2,3]) 1 2 3 > from_part([1,2]) [-1] > from_part([1,2,0]) [-1,2] > tableax([2,1]) [[1,1,2] ,[1,2,1] ] > print_tab([1,1,2]) 1 2 3 > print_tab([1,2,1]) 1 3 2 > Adams(1,adjoint) 1X[0,1,0,0] > diagram(D4) O 3 | | O---O---O 1 2 4 D4 > at = alt_tensor(2,adjoint) > at 1X[0,1,0,0] +1X[1,0,1,1] > dom_char(at) 18X[0,0,0,0] + 3X[0,0,0,2] + 3X[0,0,2,0] + 8X[0,1,0,0] + 1X[1,0,1,1] + 3X[2,0,0,0] > decomp(dom_char(at)) 1X[0,1,0,0] +1X[1,0,1,1] > W_orbit(dom_char(at)) 1X[-3, 0, 1, 1] + 1X[-3, 1,-1, 1] + 1X[-3, 1, 1,-1] + 1X[-3, 1, 1, 1] + 1X[-3, 2,-1,-1] + 1X[-3, 2,-1, 1] + 1X[-3, 2, 1,-1] + 1X[-3, 3,-1,-1] + 1X[-2,-1, 0, 2] + 1X[-2,-1, 2, 0] + 1X[-2,-1, 2, 2] + 3X[-2, 0, 0, 0] + 3X[-2, 0, 0, 2] + 3X[-2, 0, 2, 0] + 1X[-2, 1,-2, 0] + 1X[-2, 1,-2, 2] + 1X[-2, 1, 0,-2] + 8X[-2, 1, 0, 0] + 1X[-2, 1, 0, 2] + 1X[-2, 1, 2,-2] + 1X[-2, 1, 2, 0] + 3X[-2, 2,-2, 0] + 3X[-2, 2, 0,-2] + 3X[-2, 2, 0, 0] + 1X[-2, 3,-2,-2] + 1X[-2, 3,-2, 0] + 1X[-2, 3, 0,-2] + 1X[-1,-2, 1, 1] + 1X[-1,-2, 1, 3] + 1X[-1,-2, 3, 1] + 1X[-1,-1,-1, 1] + 1X[-1,-1,-1, 3] + 1X[-1,-1, 1,-1] + 8X[-1,-1, 1, 1] + 1X[-1,-1, 1, 3] + 1X[-1,-1, 3,-1] + 1X[-1,-1, 3, 1] + 1X[-1, 0,-1,-1] + 8X[-1, 0,-1, 1] + 1X[-1, 0,-1, 3] + 8X[-1, 0, 1,-1] + 8X[-1, 0, 1, 1] + 1X[-1, 0, 3,-1] + 1X[-1, 1,-3, 1] + 8X[-1, 1,-1,-1] + 8X[-1, 1,-1, 1] + 1X[-1, 1, 1,-3] + 8X[-1, 1, 1,-1] + 1X[-1, 1, 1, 1] + 1X[-1, 2,-3,-1] + 1X[-1, 2,-3, 1] + 1X[-1, 2,-1,-3] + 8X[-1, 2,-1,-1] + 1X[-1, 2,-1, 1] + 1X[-1, 2, 1,-3] + 1X[-1, 2, 1,-1] + 1X[-1, 3,-3,-1] + 1X[-1, 3,-1,-3] + 1X[-1, 3,-1,-1] + 1X[ 0,-3, 2, 2] + 3X[ 0,-2, 0, 2] + 3X[ 0,-2, 2, 0] + 3X[ 0,-2, 2, 2] + 1X[ 0,-1,-2, 2] + 8X[ 0,-1, 0, 0] + 8X[ 0,-1, 0, 2] + 1X[ 0,-1, 2,-2] + 8X[ 0,-1, 2, 0] + 1X[ 0,-1, 2, 2] + 3X[ 0, 0,-2, 0] + 3X[ 0, 0,-2, 2] + 3X[ 0, 0, 0,-2] + 18X[ 0, 0, 0, 0] + 3X[ 0, 0, 0, 2] + 3X[ 0, 0, 2,-2] + 3X[ 0, 0, 2, 0] + 1X[ 0, 1,-2,-2] + 8X[ 0, 1,-2, 0] + 1X[ 0, 1,-2, 2] + 8X[ 0, 1, 0,-2] + 8X[ 0, 1, 0, 0] + 1X[ 0, 1, 2,-2] + 3X[ 0, 2,-2,-2] + 3X[ 0, 2,-2, 0] + 3X[ 0, 2, 0,-2] + 1X[ 0, 3,-2,-2] + 1X[ 1,-3, 1, 1] + 1X[ 1,-3, 1, 3] + 1X[ 1,-3, 3, 1] + 1X[ 1,-2,-1, 1] + 1X[ 1,-2,-1, 3] + 1X[ 1,-2, 1,-1] + 8X[ 1,-2, 1, 1] + 1X[ 1,-2, 1, 3] + 1X[ 1,-2, 3,-1] + 1X[ 1,-2, 3, 1] + 1X[ 1,-1,-1,-1] + 8X[ 1,-1,-1, 1] + 1X[ 1,-1,-1, 3] + 8X[ 1,-1, 1,-1] + 8X[ 1,-1, 1, 1] + 1X[ 1,-1, 3,-1] + 1X[ 1, 0,-3, 1] + 8X[ 1, 0,-1,-1] + 8X[ 1, 0,-1, 1] + 1X[ 1, 0, 1,-3] + 8X[ 1, 0, 1,-1] + 1X[ 1, 0, 1, 1] + 1X[ 1, 1,-3,-1] + 1X[ 1, 1,-3, 1] + 1X[ 1, 1,-1,-3] + 8X[ 1, 1,-1,-1] + 1X[ 1, 1,-1, 1] + 1X[ 1, 1, 1,-3] + 1X[ 1, 1, 1,-1] + 1X[ 1, 2,-3,-1] + 1X[ 1, 2,-1,-3] + 1X[ 1, 2,-1,-1] + 1X[ 2,-3, 0, 2] + 1X[ 2,-3, 2, 0] + 1X[ 2,-3, 2, 2] + 3X[ 2,-2, 0, 0] + 3X[ 2,-2, 0, 2] + 3X[ 2,-2, 2, 0] + 1X[ 2,-1,-2, 0] + 1X[ 2,-1,-2, 2] + 1X[ 2,-1, 0,-2] + 8X[ 2,-1, 0, 0] + 1X[ 2,-1, 0, 2] + 1X[ 2,-1, 2,-2] + 1X[ 2,-1, 2, 0] + 3X[ 2, 0,-2, 0] + 3X[ 2, 0, 0,-2] + 3X[ 2, 0, 0, 0] + 1X[ 2, 1,-2,-2] + 1X[ 2, 1,-2, 0] + 1X[ 2, 1, 0,-2] + 1X[ 3,-3, 1, 1] + 1X[ 3,-2,-1, 1] + 1X[ 3,-2, 1,-1] + 1X[ 3,-2, 1, 1] + 1X[ 3,-1,-1,-1] + 1X[ 3,-1,-1, 1] + 1X[ 3,-1, 1,-1] + 1X[ 3, 0,-1,-1] > diagram(D4) O 3 | | O---O---O 1 2 4 D4 > rts = [[1,0,0,0],[0,1,0,0],[0,0,0,1]] > Cartan_type(rts) A3T1 > rtsc = closure(rts) > rtsc [[0,0,0,1] ,[0,1,0,0] ,[1,0,0,0] ] > m = res_mat(rtsc) > m [[0,0,1,1] ,[0,1,0,2] ,[0,0,0,2] ,[1,0,0,1] ] > branch(adjoint,A3T1, m) 1X[0,0,0, 0] +1X[0,1,0,-2] +1X[0,1,0, 2] +1X[1,0,1, 0] > m2=[[0,0,1,0],[0,1,0,0],[0,0,0,0],[1,0,0,0]] > branch(adjoint,A3T1,m2) 1X[0,0,0,0] +2X[0,1,0,0] +1X[1,0,1,0] > Cartan_type([[1,1,1,0],[1,1,0,0]]) A2T2 > m = res_mat([[1,1,1,0],[1,1,0,0]]) > m [[1,1,1, 0] ,[1,1,0,-2] ,[1,0,0,-1] ,[0,0,1,-1] ] > branch(adjoint,A2T2,m) Non-virtual decomposition failed. (in branch)