Decomposition Matrices

These functions interface with Thomas Breuer's package GenericDecMats

Chevie.Chars.decomposition_matrixFunction

decomposition_matrix(W,p)

This provides an interface to some decomposition matrices for Weyl groups available in the Chevie library: those for E₆, E₇, E₈ for p=2,3,5,7.

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GenericDecMats.generic_decomposition_matrixFunction

generic_decomposition_matrix(W,d)

This function obtains the Φ_d-decomposition matrix for the reductive group specified by the Coxeter group or coset W from the package GenericDecMats.

julia> W=rootdatum(:psu,5)
psu₅
julia> generic_decomposition_matrix(W,13)
!!! Φ-decomposition matrices available for ²A₄: Φ₁₀ Φ₂ Φ₄ Φ₆
julia> generic_decomposition_matrix(W,10)
Φ₁₀-decomposition matrix for psu₅
      │ps 21 ps ps ps 2111 11111
──────┼──────────────────────────
2.    │ 1  .  .  .  .    .     .
²A₂:2 │ .  1  .  .  .    .     .
11.   │ .  .  1  .  .    .     .
1.1   │ 1  .  .  1  .    .     .
.2    │ .  .  .  .  1    .     .
²A₂:11│ .  1  .  .  .    1     .
.11   │ .  .  .  1  .    .     1

The matrix itself is stored in the field .scalar of the returned struct.

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Chevie.InducedDecompositionMatrixType

InducedDecompositionMatrix(R,W,d)

returns the induced from the Levi L to the reductive group W of the generic Φ_d decomposition matrix of L.

julia> W=rootdatum(:psu,6)
psu₆

julia> L=reflection_subgroup(W,[1,2,4,5])
psu₆₍₁₂₅₄₎=(A₂A₂)₍₁₂₄₃₎Φ₁

julia> InducedDecompositionMatrix(L,W,6)
Induced Φ₆-decomposition matrix from psu₆₍₁₂₅₄₎=(A₂A₂)₍₁₂₄₃₎Φ₁ to psu₆

    │ps ps A₂
────┼─────────
²A₅ │ .  .  .
.3  │ 1  .  .
3.  │ 1  .  .
.21 │ 1  1  .
1.2 │ 2  1  .
21. │ 1  1  .
2.1 │ 2  1  .
.111│ .  1  1
111.│ .  1  1
1.11│ 1  2  1
11.1│ 1  2  1

The matrix itself is stored in the field .scalar of the returned struct.

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