# A5: finite-field analogue of A3-4. G = Sp4(F_q) on V = P + P^perp, H1 = SL(P) x SL(P^perp). # Computes dim Hom_{H1}( Ind_B^G chi , 1 (x) sigma_c ) by characters (fixed flags), and compares with # the orbit prediction [c in {(chi1/chi2)^{+-1}}] + [c in {(chi1 chi2)^{+-1}}] # + [chi1=1]([chi2=c]+[chi2=c^-1]) + [chi2=1]([chi1=c]+[chi1=c^-1]). import sys, cmath, itertools q = int(sys.argv[1]) if len(sys.argv) > 1 else 11 J = [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0]] def om(x, y): return sum(x[i] * J[i][j] * y[j] for i in range(4) for j in range(4)) % q def matvec(M, v): return [sum(M[i][j] * v[j] for j in range(len(v))) % q for i in range(len(M))] def inv(a): return pow(a, q - 2, q) g = next(x for x in range(2, q) if all(pow(x, (q - 1) // p, q) != 1 for p in range(2, q) if (q - 1) % p == 0 and all(p % r for r in range(2, p)))) LOG = {pow(g, k, q): k for k in range(q - 1)} zeta = cmath.exp(2j * cmath.pi / (q - 1)) def char(a, x): return zeta ** (a * LOG[x % q]) def kernel(M): # basis of kernel of M (rows x 4) mod q M = [row[:] for row in M]; n = 4; piv = []; r = 0 for c in range(n): p = next((i for i in range(r, len(M)) if M[i][c] % q), None) if p is None: continue M[r], M[p] = M[p], M[r] iv = inv(M[r][c]); M[r] = [x * iv % q for x in M[r]] for i in range(len(M)): if i != r and M[i][c] % q: f = M[i][c]; M[i] = [(x - f * y) % q for x, y in zip(M[i], M[r])] piv.append(c); r += 1 free = [c for c in range(n) if c not in piv] basis = [] for fc in free: v = [0] * n; v[fc] = 1 for i, pc in enumerate(piv): v[pc] = (-M[i][fc]) % q basis.append(v) return basis def solve_coords(B, x): # B: list of 4 basis vectors (columns), returns coords of x A = [[B[j][i] for j in range(4)] + [x[i]] for i in range(4)] for c in range(4): p = next(i for i in range(c, 4) if A[i][c] % q) A[c], A[p] = A[p], A[c] iv = inv(A[c][c]); A[c] = [y * iv % q for y in A[c]] for i in range(4): if i != c and A[i][c]: f = A[i][c]; A[i] = [(y - f * z) % q for y, z in zip(A[i], A[c])] return [A[i][4] for i in range(4)] def lines_in(basis): k = len(basis); out = [] for coeffs in itertools.product(range(q), repeat=k): nz = next((c for c in coeffs if c), None) if nz != 1: continue out.append([sum(c * b[i] for c, b in zip(coeffs, basis)) % q for i in range(4)]) return out def fixed_flags(h): """list of (alpha, beta) over flags (l c L) fixed by h, alpha on l, beta on L/l.""" res = [] for al in range(1, q): E = kernel([[(h[i][j] - (al if i == j else 0)) % q for j in range(4)] for i in range(4)]) if not E: continue for v in lines_in(E): perp = kernel([[sum(v[i] * J[i][j] for i in range(4)) % q for j in range(4)]]) # 3-dim # basis (v, w1, w2) of perp, completed by z cand = [v] for w in perp: M = cand + [w] if len(kernel_rows(M)) == 0: cand.append(w) if len(cand) == 3: break for z in ([1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]): if len(kernel_rows(cand + [z])) == 0: B = cand + [z]; break A = [] for wj in (B[1], B[2]): co = solve_coords(B, matvec(h, wj)) assert co[3] == 0 A.append((co[1], co[2])) a11, a21 = A[0]; a12, a22 = A[1] # column j = image of w_j # eigenlines of [[a11,a12],[a21,a22]] for be in range(1, q): K = kernel2([[(a11 - be) % q, a12], [a21, (a22 - be) % q]]) if K == 2: res += [(al, be)] * (q + 1) elif K == 1: res.append((al, be)) return res def kernel_rows(vecs): # kernel of the map coefficients -> sum c_i vecs_i (linear dependence) M = [[vecs[j][i] for j in range(len(vecs))] for i in range(4)] n = len(vecs); M = [row[:] for row in M]; r = 0; piv = [] for c in range(n): p = next((i for i in range(r, 4) if M[i][c] % q), None) if p is None: continue M[r], M[p] = M[p], M[r] iv = inv(M[r][c]); M[r] = [x * iv % q for x in M[r]] for i in range(4): if i != r and M[i][c] % q: f = M[i][c]; M[i] = [(x - f * y) % q for x, y in zip(M[i], M[r])] piv.append(c); r += 1 return [c for c in range(n) if c not in piv] def kernel2(M): # dimension of kernel of 2x2 matrix a, b = M[0]; c, d = M[1] if a % q == 0 and b % q == 0 and c % q == 0 and d % q == 0: return 2 return 1 if (a * d - b * c) % q == 0 else 0 # conjugacy classes of SL2(F_q): (rep, size, kind, data) sq = {x * x % q for x in range(1, q)} eps = next(x for x in range(2, q) if x not in sq) classes = [([[1, 0], [0, 1]], 1, 'c', 1), ([[q - 1, 0], [0, q - 1]], 1, 'c', q - 1)] for s in (1, q - 1): for b in (1, eps): classes.append(([[s, s * b % q], [0, s]], (q * q - 1) // 2, 'u', s)) for t in range(q): if t in (2, q - 2): continue disc = (t * t - 4) % q rep = [[0, q - 1], [1, t]] if disc in sq: s = next(x for x in range(1, q) if (x + inv(x)) % q == t) classes.append((rep, q * (q + 1), 's', s)) else: classes.append((rep, q * (q - 1), 'e', None)) assert sum(c[1] for c in classes) == q**3 - q def sigma_char(cl, cexp): rep, size, kind, data = cl if kind == 'c': return (q + 1) * char(cexp, data) if kind == 'u': return char(cexp, data) if kind == 's': return char(cexp, data) + char(cexp, inv(data)) return 0 FF = {} for i1, c1 in enumerate(classes): for i2, c2 in enumerate(classes): h = [[0] * 4 for _ in range(4)] for i in range(2): for j in range(2): h[i][j] = c1[0][i][j]; h[2 + i][2 + j] = c2[0][i][j] FF[(i1, i2)] = fixed_flags(h) nflags = len(FF[(0, 0)]) print("q=%d, flags fixed by identity = %d (expected %d)" % (q, nflags, (q**4 - 1) // (q - 1) * (q + 1))) def multiplicity(a1, a2, cexp): tot = 0 for (i1, i2), fl in FF.items(): if classes[i2][2] == 'e': continue chiInd = sum(char(a1, al) * char(a2, be) for al, be in fl) tot += classes[i1][1] * classes[i2][1] * chiInd * sigma_char(classes[i2], cexp).conjugate() return tot / (q**3 - q)**2 def predicted(a1, a2, k): m = q - 1; eq = lambda x, y: (x - y) % m == 0 r = (eq(k, a1 - a2) + eq(k, a2 - a1)) + (eq(k, a1 + a2) + eq(k, -a1 - a2)) r += eq(a1, 0) * (eq(a2, k) + eq(a2, -k)) + eq(a2, 0) * (eq(a1, k) + eq(a1, -k)) return r for (a1, a2) in [(1, 2), (0, 2), (1, 3), (2, 2)]: row = [] allok = True for k in range(q - 1): val = multiplicity(a1, a2, k) n = round(val.real) ok = abs(val - n) < 1e-6 and n == predicted(a1, a2, k) allok &= ok row.append(n) print("chi=(z^%d,z^%d): dim Hom for c=z^k, k=0..%d: %s ; matches orbit prediction: %s" % (a1, a2, q - 2, row, allok))