1zsy(1) User Commands zsy(1)
2
6 zsy - symmetrized tensor product of a matrix or permutation
7
9 zsy [OPTIONS] <Mode> <Inp> <Out>
10
12 This program reads a matrix or permutation, calculates its symmetrized
13 tensor product according to Mode, and writes out the result.
14 The Mode argument specifies the tensor product to be taken and the kind
16 of symmetrization to be performed. Currently there are 4 Modes avail‐
17 able:
18 s2 The symmetric tensor square. The output has size n(n+1)/2 (For
20 matrices, number of lines, for permutations, degree).
21 e2 The antisymmetric tensor square. The output has size n(n-1)/2.
23 e3 The antisymmetric tensor cube. The output has size
25 n(n-1)(n-2)/6.
26 e4 The antisymmetric fourth power. The output has size
28 n(n-1)(n-2)(n-3)/24.
29 Since the typical application of zsy is to generate new representations
31 from existing ones, it will usually be used with square matrices. How‐
32 ever, the input is not required to be square.
33 Permutations
35 Currently, only modes s2, e2 and e3 are available for permutations.
36 The result gives the operation of the input permutation on unordered
37 pairs (e2, s2) or triples (e3) of points. More precisely, if the given
38 permutation operates on 1...n, then:
39 · s2 is the operation on (i,k) with 1≤i≤k≤n.
41 · e2 is the operation on (i,k) with 1≤i<k≤n.
43 · e3 is the operation on (i,k,l) with 1≤i<k<l≤n.
45 In the output, pairs and triples are numbered lexicographically. For
47 example, E2 uses the following order: (1,2), (1,3), (2,3), (1,4), ....
48 Notice that the symmetric square is never transitive but decomposes
49 into the diagonal and the antisymmetric square. Here are some exam‐
50 ples:
51 p = (1 5 4 3 2)
53 e2(p) = (1 7 10 6 3)(2 8 4 9 5)
54 s2(p) = (1 15 10 6 3)(2 11 14 9 5)(7 14 8 4 12)
55 e3(p) = (1 5 8 10 4)(2 6 9 3 7)
56 Matrices
58 The rth exterior power (modes e2, e3, e4) has as its entries the deter‐
59 minants of r times r submatrices of the input. Rows and columns are
60 ordered lexicographically, which is equivalent to taking the following
61 basis in the tensor product:
62 e2 v_i ∧ v_j with 1≤i<j≤n
64 e3 v_i ∧ v_j ∧ v_k with 1≤i<j<k≤n
66 e4 v_i ∧ v_j ∧ v_k ∧ v_l with 1≤i<j<k<l≤n
68 The basis vectors are ordered lexicographically, for example (e2):
70 v_1∧v_2, v_1∧v_3, ..., v_1∧v_n, v_2∧v_3, v_2∧v_4, ..., v_3∧v_n, ...,
71 v_n-1∧v_n.
72 The symmetric square of a matrix with r rows and c columns is a matrix
74 with r(r+1)/2 rows and c(c+1)/2 columns, with entries given by the for‐
75 mulae
76 │
77 │ c(c-1)/2 c
78 ──────────┼────────────────
79 r*(r-1)/2 │ ad+bc ac
80 r │ 2ab a^2
81 where the upper left is the r(r-1)/2 by c(c-1)/2 matrix of permanents.
83 The program orders both the rows and the columns in lexicographical
84 order, i.e. v_1·v_2, v_1·v_3, ..., v_1·v_n, v_2·v_3, v_2·v_4, ...
85 v_2·v_n, v_3·v_4, ... v_{n-1}·v_n, v_1·v_1, v_2·v_2, ... v_n·v_n, with
86 the assumption that v_i·v_j = v_j·v_i, i.e. the action is on quadratic
87 polynomials.
88 The symmetric square is, in general, irreducible except in characteris‐
90 tic 2. In that case there is a copy of the Frobenius square as an
91 invariant submodule, as can be seen from the 2ab in the above formulae.
92 Invariant subspaces in characteristic 2 correspond to special groups
93 (i.e., groups of the form 2^nx2^m) on which the group given acts on the
94 quotient 2^n.
95 Here are some examples:
97 (1 2 1 3) (1 2 1 3 6 2)
99 E2 (0 1 2 1) = (0 1 0 2 0 4) (mod 7)
100 (1 2 2 3) (6 5 6 5 1 4)
101 (1 0 2 0 2) (1 0 1 4 0 1 0 0 0 0)
103 E3 (1 1 2 1 2) = (1 4 3 4 0 3 2 1 3 4) (mod 5)
104 (3 3 2 3 2) (1 2 2 3 2 3 1 3 4 2)
105 (1 2 3 1 0) (4 0 4 0 2 0 4 2 1 3)
106 (1 2 1 5 5 7 0 2 2 3)
108 (1 2 1 3) (4 3 6 6 12 9 1 4 2 9)
109 S2 (0 1 2 1) = (1 2 1 6 5 8 0 2 4 3) (mod 13)
110 (1 2 2 3) (4 2 6 4 12 6 1 4 1 9)
111 (0 0 0 4 2 4 0 1 4 1)
112 (4 4 6 8 12 12 1 4 4 9)
113
115 -Q Quiet, no messages.
116 -V Verbose, more messages.
118 -T <MaxTime>
120 Set CPU time limit
121 -G Produce output in GAP format. This option implies -Q.
123
125 If the input file contains more than one permutation, only the first
126 permutation is read in and processed.
127 If the input is a matrix, the whole input matrix and one row of the
129 result must fit into memory. In case of permutations both the input
130 and the result must fit into memory.
131MeatAxe 2.4.24 zsy(1)