From d643acabbc4d87826e3497951d239b6f72c72035 Mon Sep 17 00:00:00 2001 From: flavis Date: Sun, 22 May 2022 22:27:19 +0200 Subject: [PATCH] add lie --- sose2022/lie/sheet1.xopp | Bin 0 -> 1673428 bytes sose2022/lie/sheet3.pdf | Bin 0 -> 2023815 bytes sose2022/lie/sheet3.xopp | Bin 0 -> 2476850 bytes sose2022/lie/sheet4.py | 108 +++++++++++++++++++++++++++++++++++++++ sose2022/lie/sheet4.xopp | Bin 0 -> 1442917 bytes sose2022/lie/skript.pdf | Bin 0 -> 975377 bytes sose2022/lie/skript.xopp | Bin 0 -> 538316 bytes 7 files changed, 108 insertions(+) create mode 100644 sose2022/lie/sheet1.xopp create mode 100644 sose2022/lie/sheet3.pdf create mode 100644 sose2022/lie/sheet3.xopp create mode 100644 sose2022/lie/sheet4.py create mode 100644 sose2022/lie/sheet4.xopp create mode 100644 sose2022/lie/skript.pdf create mode 100644 sose2022/lie/skript.xopp diff --git a/sose2022/lie/sheet1.xopp b/sose2022/lie/sheet1.xopp new file mode 100644 index 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assert(len(perm) == n) + return list(map(sorted, split_rows(part, canon_tab(n)))) == list(map(sorted, split_rows(part, list(perm)))) + +def cycles(perm): + cycles = [] + xs = list(range(1, len(perm)+1)) + while len(xs) > 0: + c = [xs[0]] + k = xs[0] + while (perm[k-1] not in c): + k = perm[k-1] + c.append(k) + xs = [ x for x in xs if x not in c ] + cycles.append(c) + return cycles + +def cycle_type(perm): + cs = cycles(perm) + types = {} + for c in cs: + if len(c) not in types: + types[len(c)] = 1 + else: + types[len(c)] += 1 + t = [] + for i in range(1, max(perm)+1): + t.append(types.get(i, 0)) + return tuple(t) + +def conjugacy_classes(n): + cs = {} + for perm in s_n(n): + ct = cycle_type(perm) + if ct not in cs: + cs[ct] = [perm] + else: + cs[ct].append(perm) + l = list(cs.items()) + return sorted(l, key=lambda x: x[0], reverse=True) + +def partitions(n, conj_classes=None): + parts = [] + if conj_classes == None: + conj_classes = conjugacy_classes(n) + for cycle_type, conj in conj_classes: + part = [] + for i, c in enumerate(cycle_type): + for s in range(0, c): + part.append(i+1) + parts.append(sorted(part, reverse=True)) + #parts.append([(i+1) * cycle_type[i] for i in range(0, n)]) + return list(sorted(parts, reverse=True)) + +def psi(n): + cs = conjugacy_classes(n) + ps = partitions(n, cs) + m = np.zeros((len(cs), len(cs))) + for i, part in enumerate(ps): + row_stabs = set(filter(lambda sigma: is_row_stab(part, sigma), s_n(n))) + for j, (cycle_type, conj) in enumerate(cs): + m[i, j] = len(row_stabs & set(conj)) * np.math.factorial(n) / (len(row_stabs) * len(conj)) + return m + +def sigma(n): + cs = conjugacy_classes(n) + m = np.zeros((len(cs), len(cs))) + for i, (cycle_type, conj) in enumerate(cs): + m[i, i] = len(conj) / np.math.factorial(n) + return m + +if __name__ == '__main__': + if len(sys.argv) >= 2: + n = int(sys.argv[1]) + else: + n = 3 + + psi = psi(n) + sigma = sigma(n) + + a = np.dot(psi, np.dot(sigma, np.transpose(psi))) + k = 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