[LiePRing] 5 Advanced functions for Lie p-rings

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# 5 Advanced functions for Lie p-rings

### Sections

1. Schur multipliers
2. Automorphism groups

This chapter described a few more advanced functions available for generic Lie p-rings.

## 5.1 Schur multipliers

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.

- `LiePSchurMult( L )`

  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'.

      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 := [  ] ) ]

  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 p and of order p2. The second option only arises if xy = 0 and thus, as x is non-zero, if y = 0.

  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 p in this case. If it does not succeed then it returns fail.

- `ElementNumbers( pp, ss )`

  We continue the above example.

      gap> ElementNumbers(ParametersOfLiePRing(L), ss);
      rec( norms := [ [ p^2 ], [ p ] ], numbs := [ p-1, p^2-2*p+1 ] )

  ## 5.2 Automorphism groups

  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.

- `AutGrpDescription( L )`

  Each automorphism of L is defined by its images on a generating set of L. If l1, ·.·, ln is a basis of L and l1, ·., ld is a generating set, then each automorphism is defined by the images of l1, ·., ld and each image is an integral linear combination of the basis elements l1, ·., ln. 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.

      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 ] ] )

  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 p10 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 A222 = 1 and A24 = A22 A232 / 2. Hence L has 2 p9 automorphisms.

  The entry eqns is a list of lists. The equations in the ith entry of this list have to be satisfied mod pi.

  In a few special cases, the function returns a list of possible automorphisms together with related equations and conditions. We exhibit an example.

      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 ] ] ) ]

  In this example A12 x = −1 modulo p2. We note that different choices for A12 do not give different automorphisms. Hence a single solution for A12 is sufficient to describe all automorphisms.

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