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Unitary matrices over finite fields

asked 2012-12-30 05:58:09 -0600

GaryMak gravatar image

updated 2012-12-31 01:40:47 -0600

Hi everyone

I was delighted to see that the SAGE team had implemented the unitary groups GU(n,q) and SU(n,q), since they are such peculiar objects; however the implementation does not seem to include many matrix functions (eg det, trace, transpose etc.).

Let S be a (small) subset of GF(q^2). Let G=GU(n,q) and define a subset D_S of G (ie D_S is NOT a vector subspace or subgroup or anything) to be the matrices whose entries may only be chosen from this particular subset S. Let U, V in D_S. I need to check the matrix products (U^t)V for membership of D_S, where U^t denotes U.transpose() acted on by Frobenius in the usual way, and ideally I would like to store the set of pairs P = {(U,V) satisfying (U^t)V in D_S}, or even better to store some "generators" for P.

I have two questions:

(1) which matrix operations are available for elements of GU(n,q)?

(2) is there an easy way of specifying a sub-SET like D_S so that such a search may be performed? Needless to say once q and/or n gets any bigger than 3, holding any of these subsets naively in memory becomes prohibitively costly!

Many thanks for any help.

PS I can find generators for these groups using G.gens(); how please do I find relations between the gens?

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answered 2017-03-01 16:52:53 -0600

dan_fulea gravatar image

The setting is clear, but the question somehow interferes with the mathematical research question of finding a good $S$, subset in $\mathbb{F}_{p^2}$, $p$ prime.

My structural problem is that $S$ should be good for all entries in a matrix to be validated. The property is defined elementwise, so we need the matrix with its entries. (Some triangular or "parabolic" shapes are eliminated.)

Now to the questions:

(1) which matrix operations are available for elements of GU(n,q)?

I tried:

sage: G = GU( 3, 7 )
sage: G
General Unitary Group of degree 3 over Finite Field in a of size 7^2
sage: G.order().factor()
2^10 * 3 * 7^3 * 43
sage: A = G.random_element()
sage: A
[  a + 1       5 4*a + 1]
[    6*a 6*a + 3     6*a]
[5*a + 6 2*a + 1 3*a + 1]
sage: A.
A.N                     A.conjugacy_class       A.inverse               A.matrix                A.parent                A.save                  
A.base_extend           A.db                    A.is_idempotent         A.multiplicative_order  A.powers                A.subs                  
A.base_ring             A.dump                  A.is_one                A.n                     A.pseudo_order          A.substitute            
A.cartesian_product     A.dumps                 A.is_zero               A.numerical_approx      A.rename                A.word_problem          
A.category              A.gap                   A.list                  A.order                 A.reset_name

Above, after A. i was hitting the TAB (twice), and the list of implemented (public) methods was printed. An other way of investigating A is by printing dir(A). The "hidden" methods show themselves after A._TABULATOR . (In the interpreter.) The we can use for instance the method order.

sage: A.order().factor()
2^3 * 43

Yes, determinant is not in the landscape. I tried to coerce, to apply a morphism. Finally, the following work around could indeed walk around.

p = 7
G = GU( 3, p )

F.<a> = GF( p^2 )
H = GL( 3, F )

H_Matrices = []    # and we will append.
for A in G.gens():
    HA = H ( A.gap().sage() )    # tricky coercion, not proud of it
    H_Matrices.append( HA )

HG = H.subgroup( H_Matrices )
print "HG has order: %s" % HG.order().factor()

And we get:

HG has order: 2^10 * 3 * 7^3 * 43

And in HG we also have the determinant. For instance:

h = HG.random_element()
print "The matrix\n%s\nhas determinant %s" % ( h, h.matrix().det() )

This gives a value in $\pm 1$:

The matrix
[      a 6*a + 1 4*a + 1]
[5*a + 5     3*a 2*a + 2]
[2*a + 3 3*a + 3   a + 3]
has determinant 6

(2) For the second question, it is easy to implement a validator. For instance:

A, B = H_Matrices
A = A.matrix()
B = B.matrix()

S = set( A.list() + B.list() )
# S is the list of entries in A, B
FA = H( [ F.frobenius_endomorphism()( entry ) for entry in A.list() ] ).matrix()
FB = H( [ F.frobenius_endomorphism()( entry ) for entry in B.list() ] ).matrix()

So we obtain the $F'$ action on matrices, which invokes the Frobenius $F$ and the reversion of rows and columns, that invariates $A$.

Now we can build products, and check if the entries land in S.

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Asked: 2012-12-30 05:58:09 -0600

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Last updated: Mar 01