GroupRepresentation(G, M, action) : GrpRed, CombFreeMod, MonStgElt -> ModRed
params: List Default: [* *]
Constructs a group representation for the group G on the combinatorial free module
M with basis B, such that the action on basis elements G x B to M is described
by the map given by the string action.
SubCFModule(V, t) : CombFreeMod, Any -> CombFreeMod, CombFreeModHom
The subrepresentation (or submodule) of V whose underlying free module
is generated by t.
TrivialRepresentation(G, R) : GrpRed, Rng -> ModRed
name: MonStgElt Default: "v"
The trivial representation for the group G over the ring R,
where the basis element has name name.
StandardRepresentation(G) : GrpRed -> ModRed
name: MonStgElt Default: "x"
The standard representation of the matrix group G over its ring of definition R,
i.e. the representation obtained by considering its given embedding in GLn(R) acting on Rn
by invertible linear transformations.
The basis will have names x1, ..., xn, where
x is specified by name.
name: MonStgElt Default: "x"
The 1-dimensional representation spind
of the orthogonal group G
induced by the spinor norm and d.
The representation detk tensor Symj.
The representation spind tensor Symk
of the orthogonal group G.
The representation spind tensor Altj
of the orthogonal group G.
The character θp from [DPRT24].
If G = OO(Q) is an orthogonal group defined over Q,
with Q integral, and p is a prime divisor of Disc(Q),
returns the 1-dimensional representation of G(Zp)
on the determinant of the radical of Q mod 2p.
The character θd = ∏p | d θp, where θp
is the character constructed by the function RadicalSignCharacterSinglePrime.
The spin representation of the special orthogonal group
G=SO(Q) with coefficients in Fp.
k: RngIntElt Default: 1
name: MonStgElt Default: "v"
The 1-dimensional representation, where G acts
via the g |-> det(g)k, where the basis element has name name.
The symmetric representation Symn(V).
The alternating representation Altn(V).
The dual (contragredient) representation Vv.
The representation V tensor W with the diagonal action.
The tensor power representation V tensor d,
with the diagonal action.
If f : G to H is a group homomorphism,
and V is a representation of H,
returns the pullback of V via f to
a representation of G.
Does not verify that f is a group homomorphism.
AlternatingPower(M, n) : CombFreeMod, RngIntElt -> CombFreeMod
Returns bigwedgen M.
The underlying module of the exterior algebra of M, namely
bigwedge M,
together with an embedding of M as the degree 1 component.
The direct sum bigoplusi Mi.
The algebraic representation of the group G with highest weight w.
Embeds the group of Lie type G into its standard representation.
The irreducible algebraic representation of G with highest weight w.
If G is a reductive group defined over a number field F,
return the irreducible algebraic representation of G with highest weight w,
and coefficient field of characteristic p, obtained by
the reduction modulo a prime P of F above p.
CombinatorialFreeModule(R, S) : Rng, [ MonStgElt ] -> CombFreeMod
params: List Default: [* *]
A combinatorial free module over the ring R with basis given by S,
namely M = RS.
V2.29, 21 October 2025