71.16 IsNewIndecomposable

IsNewIndecomposable(d, x [,mu])

IsNewIndecomposable is the function which does all of the hard work when the function InducedDecompositionMatrix is applied to decomposition matrices (see InducedDecompositionMatrix). Given a projective module x, IsNewIndecomposable returns true if it is able to show that x is indecomposable (and this indecomposable is not already listed in d), and false otherwise. IsNewIndecomposable will also print a brief description of its findings, giving an upper and lower bound on the first decomposition number mu for which it is unable to determine the multiplicity of S(mu) in x.

IsNewIndecomposable works by running through all of the partitions nu such that P(nu) could be a summand of x and it uses various results, such as the q-Schaper theorem of [JM2] (see Schaper Schaper), the Mullineux map (see Mullineux Mullineux), and inducing simple modules, to determine if P(nu) does indeed split off. In addition, if d is the decomposition matrix for 'H'(Sym_n) then IsNewIndecomposable will probably use some of the decomposition matrices of 'H'(Sym_m) for m le n, if they are known. Consequently it is a good idea to save decomposition matrices as they are calculated (see SaveDecompositionMatrix).

For example, in calculating the 2--modular decomposition matrices of Sym_{r} the first projective which InducedDecompositionMatrix is unable to calculate is P(10).

gap> H:=Specht(2,2);; 
gap> d:=InducedDecompositionMatrix(DecompositionMatrix(H,9));; 
# Inducing.
# The following projectives are missing from <d>:
#  [ 10 ]

(In fact, given the above commands, Specht will return the full decomposition matrix for Sym_{10} because this matrix is in the library; these were the commands that I used to calculate the decomposition matrix in the library.)

By inducing P(9) we can find a projective H--module which contains P(10). We can then use IsNewIndecomposable to try and decompose this induced module into a sum of PIMs.

gap> SpechtPrettyPrint(); x:=InducedModule(H.P(9),1);
S(10)+S(9,1)+S(8,2)+2S(8,1^2)+S(7,3)+2S(7,1^3)+3S(6,3,1)+3S(6,2^2)
+4S(6,2,1^2)+2S(6,1^4)+4S(5,3,2)+5S(5,3,1^2)+5S(5,2^2,1)+2S(5,1^5)
+2S(4^2,2)+2S(4^2,1^2)+2S(4,3^2)+5S(4,3,1^3)+2S(4,2^3)+5S(4,2^2,1^2)
+4S(4,2,1^4)+2S(4,1^6)+2S(3^3,1)+2S(3^2,2^2)+4S(3^2,2,1^2)
+3S(3^2,1^4)+3S(3,2^2,1^3)+2S(3,1^7)+S(2^3,1^4)+S(2^2,1^6)+S(2,1^8)
+S(1^10)
gap> IsNewIndecomposable(d,x);
# The multiplicity of S(6,3,1) in P(10) is at least 1 and at most 2.
false
gap> x;
S(10)+S(9,1)+S(8,2)+2S(8,1^2)+S(7,3)+2S(7,1^3)+2S(6,3,1)+2S(6,2^2)
+3S(6,2,1^2)+2S(6,1^4)+3S(5,3,2)+4S(5,3,1^2)+4S(5,2^2,1)+2S(5,1^5)
+2S(4^2,2)+2S(4^2,1^2)+2S(4,3^2)+4S(4,3,1^3)+2S(4,2^3)+4S(4,2^2,1^2)
+3S(4,2,1^4)+2S(4,1^6)+2S(3^3,1)+2S(3^2,2^2)+3S(3^2,2,1^2)
+2S(3^2,1^4)+2S(3,2^2,1^3)+2S(3,1^7)+S(2^3,1^4)+S(2^2,1^6)+S(2,1^8)
+S(1^10)

Notice that some of the coefficients of the Specht modules in x have changed; this is because IsNewIndecomposable was able to determine that the multiplicity of S(6,3,1) was at most 2 and so it subtracted one copy of P(6,3,1) from x.

In this case, the multiplicity of S(6,3,1) in P(10) is easy to resolve because general theory says that this multiplicity must be odd. Therefore, x-'P'(6,3,1) is projective. After subtracting P(6,3,1) from x we again use IsNewIndecomposable to see if x is now indecomposable. We can tell IsNewIndecomposable that all of the multiplicities up to and including S(6,3,1) have already been checked by giving it the addition argument mu=[6,3,1].

gap> x:=x-H.P(d,6,3,1);; IsNewIndecomposable(d,x,6,3,1);
true

Consequently, <x>='P'(10) and we add it to the decomposition matrix d (and save it).

gap> AddIndecomposable(d,x); SaveDecompositionMatrix(d);

A full description of what IsNewIndecomposable does can be found by reading the comments in specht.g. Any suggestions or improvements on this function would be especially welcome.

See also DecompositionMatrix DecompositionMatrix and InducedDecompositionMatrix InducedDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

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GAP 3.4.4
April 1997