# S1-4 checks for GL_{2m}/Sp_{2m}: # (a) dim_k {X in gl_2m(k[t]/t^N) : X D_a + D_a X^T = 0} = (2m^2+m)N + sum_j (4j-3) a_j, # by linear algebra over F_p (p = 3, 5, 7), all labels a >= 0 with a_1 < N, m = 1..3. # (b) Cayley transform: for random X in that space with 1-X invertible, # g = (1-X)^{-1}(1+X) satisfies g D_a g^T = D_a mod t^N. # (c) def(nu) = n_1(nu) for nu in {0,1,2}^m, |nu| = k, m <= 8, using only the orbit-dimension formula. import itertools, random, sys random.seed(1) def labels(m, N): for a in itertools.product(range(N), repeat=m): if all(a[i] >= a[i+1] for i in range(m-1)): yield a # matrices over R = F_p[t]/t^N : entries are lists of length N def Dmat(a, N, p): n = 2*len(a) D = [[[0]*N for _ in range(n)] for _ in range(n)] for j, aj in enumerate(a): if aj < N: D[2*j][2*j+1][aj] = 1 D[2*j+1][2*j][aj] = p-1 return D def rmul(x, y, N, p): r = [0]*N for i, c in enumerate(x): if c: for j in range(N-i): if y[j]: r[i+j] = (r[i+j] + c*y[j]) % p return r def mmul(A, B, N, p): n = len(A) C = [[[0]*N for _ in range(n)] for _ in range(n)] for i in range(n): for k in range(n): if any(A[i][k]): for j in range(n): if any(B[k][j]): pr = rmul(A[i][k], B[k][j], N, p) C[i][j] = [(u+v) % p for u, v in zip(C[i][j], pr)] return C def transpose(A): n = len(A) return [[A[j][i] for j in range(n)] for i in range(n)] def rank_mod(rows, p): rows = [r[:] for r in rows] rk = 0; ncol = len(rows[0]) if rows else 0 for c in range(ncol): piv = None for i in range(rk, len(rows)): if rows[i][c] % p: piv = i; break if piv is None: continue rows[rk], rows[piv] = rows[piv], rows[rk] inv = pow(rows[rk][c], p-2, p) rows[rk] = [(v*inv) % p for v in rows[rk]] for i in range(len(rows)): if i != rk and rows[i][c] % p: f = rows[i][c] rows[i] = [(u - f*v) % p for u, v in zip(rows[i], rows[rk])] rk += 1 return rk, rows def lie_dim(a, N, p): n = 2*len(a); D = Dmat(a, N, p) nv = n*n*N # linear map X -> X D + D X^T ; columns indexed by (i,j,e) basis matrices t^e E_ij cols = [] for i in range(n): for j in range(n): for e in range(N): X = [[[0]*N for _ in range(n)] for _ in range(n)] X[i][j][e] = 1 Y = mmul(X, D, N, p); Z = mmul(D, transpose(X), N, p) vec = [] for r in range(n): for s in range(n): vec += [(u+v) % p for u, v in zip(Y[r][s], Z[r][s])] cols.append(vec) # rank of the map = rank of matrix with these columns rk, _ = rank_mod(cols, p) return nv - rk, cols def formula(a, N): m = len(a) return (2*m*m+m)*N + sum((4*(j+1)-3)*aj for j, aj in enumerate(a)) ok = True for p in [3, 5, 7]: for m in [1, 2, 3]: for N in ([2, 3, 4] if m < 3 else [2, 3]): if p == 7 and m == 3 and N == 3: continue for a in labels(m, N): dL, _ = lie_dim(a, N, p) if dL != formula(a, N): ok = False; print('MISMATCH', p, m, N, a, dL, formula(a, N)) print('p=%d: Lie-algebra dimension formula checked, all ok so far: %s' % (p, ok)); sys.stdout.flush() # (b) Cayley transform def kernel_basis(a, N, p): n = 2*len(a); D = Dmat(a, N, p) _, cols = lie_dim(a, N, p) # solve sum z_k cols[k] = 0 : build matrix with rows = equations neq = len(cols[0]); nv = len(cols) A = [[cols[k][r] for k in range(nv)] for r in range(neq)] rk, R = rank_mod(A, p) piv = [] for r in range(rk): c = next(c for c in range(nv) if R[r][c] % p) piv.append(c) free = [c for c in range(nv) if c not in piv] basis = [] for f in free: z = [0]*nv; z[f] = 1 for r, c in enumerate(piv): z[c] = (-R[r][f]) % p basis.append(z) return basis def vec_to_mat(z, n, N): X = [[[0]*N for _ in range(n)] for _ in range(n)] k = 0 for i in range(n): for j in range(n): for e in range(N): X[i][j][e] = z[k]; k += 1 return X def inv_mat(A, N, p): # Gauss-Jordan over R = F_p[t]/t^N for A with det unit n = len(A) M = [[A[i][j][:] for j in range(n)] + [([1]+[0]*(N-1)) if i == j else [0]*N for j in range(n)] for i in range(n)] def runit_inv(x): g = [0]*N; g[0] = pow(x[0], p-2, p) for k in range(1, N): s = sum(x[i]*g[k-i] for i in range(1, k+1)) g[k] = (-s*g[0]) % p return g for c in range(n): piv = next(i for i in range(c, n) if M[i][c][0] % p) M[c], M[piv] = M[piv], M[c] u = runit_inv(M[c][c]) M[c] = [rmul(u, x, N, p) for x in M[c]] for i in range(n): if i != c and any(M[i][c]): f = M[i][c] M[i] = [[(x - y) % p for x, y in zip(M[i][j], rmul(f, M[c][j], N, p))] for j in range(2*n)] return [row[n:] for row in M] cay_ok = True; tested = 0 for p in [3, 5]: for (m, N, a) in [(2, 3, (2, 0)), (2, 3, (1, 1)), (3, 3, (2, 1, 0)), (3, 4, (3, 1, 1))]: n = 2*m; D = Dmat(a, N, p); B = kernel_basis(a, N, p) for trial in range(5): z = [0]*len(B[0]) for b in B: c = random.randrange(p) z = [(u + c*v) % p for u, v in zip(z, b)] X = vec_to_mat(z, n, N) I = [[([1]+[0]*(N-1)) if i == j else [0]*N for j in range(n)] for i in range(n)] OmX = [[[(u - v) % p for u, v in zip(I[i][j], X[i][j])] for j in range(n)] for i in range(n)] OpX = [[[(u + v) % p for u, v in zip(I[i][j], X[i][j])] for j in range(n)] for i in range(n)] try: g = mmul(inv_mat(OmX, N, p), OpX, N, p) except StopIteration: continue lhs = mmul(mmul(g, D, N, p), transpose(g), N, p) tested += 1 if lhs != D: cay_ok = False; print('Cayley FAIL', p, a) print('(b) Cayley transform lands in the stabilizer: %s (%d random tests)' % (cay_ok, tested)) # (c) defect formula def dimO(nu): # orbit dimension minus m(2m-1)N return -sum((4*(j+1)-3)*v for j, v in enumerate(nu)) def_ok = True for m in range(1, 9): for k in range(1, 2*m): nus = [nu for nu in itertools.product([2, 1, 0], repeat=m) if sum(nu) == k and all(nu[i] >= nu[i+1] for i in range(m-1))] eps = k % 2 top = max(nus, key=dimO) for nu in nus: n1 = nu.count(1) c = eps + dimO(top) - dimO(nu) d = n1*(n1+1) - c if d != n1: def_ok = False; print('defect mismatch', m, k, nu, d, n1) print('(c) def(nu) = n_1(nu) for m <= 8, 1 <= k <= 2m-1: %s' % def_ok)