<html><head><title>[LiePRing] 5 Advanced functions for Lie p-rings</title></head>
<body text="#000000" bgcolor="#ffffff">
[<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href = "theindex.htm">Index</a>]
<h1>5 Advanced functions for Lie p-rings</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP005.htm#SECT001">Schur multipliers</a>
<li> <A HREF="CHAP005.htm#SECT002">Automorphism groups</a>
</ol><p>
<p>
This chapter described a few more advanced functions available for
generic Lie p-rings.
<p>
<p>
<h2><a name="SECT001">5.1 Schur multipliers</a></h2>
<p><p>
The package contains a method to determine the Schur multipliers of the
Lie p-rings in the family defined by a generic Lie p-ring.
<p>
<a name = "SSEC001.1"></a>
<li><code>LiePSchurMult( L )</code>
<p>
The function takes as input a generic Lie p-ring and determines a list
of possible Schur multipliers, each described by its abelian invariants,
for the Lie p-rings in the family described by L. For each entry in the
list of Schur multipliers there is a description of those parameters which
give the considered entry. This description consists of two lists 'units' and 'zeros'. Both consist of rational functions over the parameters
of the Lie p-ring. The parameters described by these lists are which evaluate
to zero for each rational function in 'zeros' and evaluate not to zero
for each rational function in 'units'.
<p>
<pre>
gap> LL := LiePRingsByLibrary(7);;
gap> L := Filtered(LL, x -> Length(ParametersOfLiePRing(x))=2)[1];
<LiePRing of dimension 7 over prime p with parameters [ x, y ]>
gap> NumberOfLiePRingsInFamily(L);
p^2-p
gap> RingInvariants(L);
rec( units := [ x ], zeros := [ ] )
gap> ss := LiePSchurMult(L);
[ rec( norm := [ p ], units := [ x, y ], zeros := [ x*y^2-x*y+1 ] ),
rec( norm := [ p^2 ], units := [ x ], zeros := [ x*y ] ),
rec( norm := [ p ], units := [ x, x*y^2-x*y+1, y ], zeros := [ ] ) ]
</pre>
<p>
In this example, L defines a generic Lie p-rings with two parameters and
the RingInvariants of L show that the parameter x should be non-zero. The
function LiePSchurMult(L) yields that there are two possible Schur
multipliers for the Lie p-rings in the family defined by L: the cyclic
groups of order <i>p</i> and of order <i>p</i><sup>2</sup>. The second option only arises
if <i>xy</i> = 0 and thus, as <i>x</i> is non-zero, if <i>y</i> = 0.
<p>
The package also contains a function that tries to determine the numbers of
values of the parameters satisfying the conditions of a description of
a Schur multiplier. This succeeds in many cases and returns a polynomial
in <i>p</i> in this case. If it does not succeed then it returns fail.
<p>
<a name = "SSEC001.2"></a>
<li><code>ElementNumbers( pp, ss )</code>
<p>
We continue the above example.
<p>
<pre>
gap> ElementNumbers(ParametersOfLiePRing(L), ss);
rec( norms := [ [ p^2 ], [ p ] ], numbs := [ p-1, p^2-2*p+1 ] )
</pre>
<p>
<p>
<h2><a name="SECT002">5.2 Automorphism groups</a></h2>
<p><p>
The package contains a function that determines a description for the
automorphism groups of the Lie p-rings in the family defined by a generic
Lie p-ring.
<p>
<a name = "SSEC002.1"></a>
<li><code>AutGrpDescription( L )</code>
<p>
Each automorphism of L is defined by its images on a generating
set of L. If <i>l</i><sub>1</sub>, ·.·, <i>l</i><sub><i>n</i></sub> is a basis of L and <i>l</i><sub>1</sub>, ·., <i>l</i><sub><i>d</i></sub> is a
generating set, then each automorphism is defined by the images of
<i>l</i><sub>1</sub>, ·., <i>l</i><sub><i>d</i></sub> and each image is an integral linear combination of
the basis elements
<i>l</i><sub>1</sub>, ·., <i>l</i><sub><i>n</i></sub>. The function AutGrpDescription returns a matrix containing
a description of the coefficients in each linear combination and a list
of relations among these coefficients. We consider two examples.
<p>
<pre>
gap> L := Filtered(LL, x -> Length(ParametersOfLiePRing(x))=2)[1];
<LiePRing of dimension 7 over prime p with parameters [ x, y ]>
gap> AutGroupDescription(L);
rec( auto := [ [ 1, 0, A13, A14, A15, A16, A17 ],
[ 0, 1, A23, A24, A25, A26, A27 ] ],
eqns := [ [ ], [ ] ] )
gap> L := Filtered(LL, x -> Length(ParametersOfLiePRing(x))=2)[2];
<LiePRing of dimension 7 over prime p with parameters [ x, y ]>
gap> AutGroupDescription(L);
rec( auto := [ [ A22^3, 0, A13, A14, A15, A16, A17 ],
[ 0, A22, A23, A24, A25, A26, A27 ] ],
eqns := [ [ A22*A24-1/2*A23^2, A22^2*y-y,
A22*A23^2*y-2*A24*y, A22^4-1,
A23^4*y-4*A24^2*y, A22^3*A23^2-2*A24,
A22^2*A23^4-4*A24^2, A22*A23^6-8*A24^3,
A23^8-16*A24^4 ] ] )
</pre>
<p>
In both cases, L is generated by the first two entries in its
basis and hence the automorphism group matrix has two rows and seven
columns. In the first case, L has <i>p</i><sup>10</sup> automorphisms inducing the
identity on the Frattini-quotient of L. In the second case, the automorphism
group matrix shows that each automorphism induces a certain type of
diagonal matrix on the Frattini-quotient of L and there are further
equations among the coefficients of the matrix. These further equations
are equivalent to <i>A</i>22<sup>2</sup> = 1 and <i>A</i>24 = <i>A</i>22 <i>A</i>23<sup>2</sup> / 2. Hence L has
2 <i>p</i><sup>9</sup> automorphisms.
<p>
The entry eqns is a list of lists. The equations in the ith entry of
this list have to be satisfied mod <i>p</i><sup><i>i</i></sup>.
<p>
In a few special cases, the function returns a list of possible
automorphisms together with related equations and conditions.
We exhibit an example.
<p>
<pre>
gap> L := LiePRingsByLibrary(7)[489];
<LiePRing of dimension 7 over prime p with parameters [ x ]>
gap> AutGroupDescription(L);
[ rec( auto := [ [ 1, 0, A13, A14, A15, A16, A17 ],
[ 0, 1, A23, A24, A25, A26, A27 ] ],
comment := "p^8 automorphisms",
eqns := [ [ A13^2*x-A13*A23+2*A15*x+A14-A25,
-A13*A23*x+A14*x+A23^2-A25*x-2*A24 ] ] ),
rec( auto := [ [ 0, A12, A13, A14, A15, A16, A17 ],
[ -x, 0, A23, A24, A25, A26, A27 ] ],
comment := "p^8 automorphisms when x <> 0 mod p",
eqns := [ [ A12^2*A24*x-A12*A13*A23*x+A12*A13*x^2
+2*A12*A15*x^2+A12*A14*x-A13^2*x+A13*x+A15*x-A14,
-A12^2*A23*x^3+A12*A13*x^3+A12*A23^2*x-A12*A25*x^2
-2*A12*A24*x+A13*A23*x+A13*x^2-A15*x^2+A23*x+A25*x-A24 ],
[ A12*x+1 ] ] ) ]
</pre>
<p>
In this example <i>A</i>12 <i>x</i> = −1 modulo <i>p</i><sup>2</sup>. We note that different choices
for <i>A</i>12 do not give different automorphisms. Hence a single solution for
<i>A</i>12 is sufficient to describe all automorphisms.
<p>
<p>
[<a href = "chapters.htm">Up</a>] [<a href ="CHAP004.htm">Previous</a>] [<a href = "theindex.htm">Index</a>]
<P>
<address>LiePRing manual<br>June 2024
</address></body></html>
Messung V0.5
¤ Dauer der Verarbeitung: 0.1 Sekunden
(vorverarbeitet)
¤
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.
