Symbols

Chevie.Symbols — Module

The combinatorial objects in this module are partitions, β-sets and symbols.

A partition is a non-increasing list of positive integers p₁≥p₂≥…pₙ≥0, represented as a Vector{Int}, which is normalized if it has no trailing zeroes.

A β-set is a strictly increasing Vector of nonnegative integers, up to shift, the equivalence relation generated by the elementary shifts [b₁,…,bₙ]∼[0,1+b₁,…,1+bₙ]. An equivalence class has exactly one member which does not contain 0: it is called a normalized β-set.

To a normalized partition p₁≥p₂≥…pₙ>0 is associated a β-set, whose normalized representative is pₙ,pₙ₋₁+1,…,p₁+n-1. A mnemonic is that this is the areas of the hooks in the Young diagram going up along the first column. Conversely, to each β-set b₁<b₂<…<bₙ is associated the partition bₙ-n+1≥…≥b₂-1≥b₁, which may have some trailing zeros if starting from a non-normalized representative.

The functions for βsets in this module are

βset which constructs a normalized βset from a partition.
shiftβ which shifts a βset
partβ which constructs a partition from a βset

As a generalisation of β-sets, Lusztig1977 has introduced 2-symbols and more general e-symbols were introduced in Malle1995. An e-symbol is a vector S=[S₁,…,Sₑ] of β-sets, taken modulo the equivalence relation generated by the simultaneous elementary shift of all β-sets, and by cyclic permutations of S; in the particular case where e=2, S is thus an unordered pair of β-sets. S is a normalized symbol if 0 is not in the intersection of the Sᵢ; equivalent normalized symbols are equivalent by cyclic permutation. The content of S is mod(c,e) where c=sum(length.(S)); it is an invariant of the symbol, as well as the rank, defined for an e-symbol as sum(sum,S)-div((c-1)*(c-e+1),2*e). Invariant by shift but not cyclic permutation is the shape s.-minimum(s) where s=length.(S).

When e=2 we choose representatives of the symbols [S₁,S₂] such that length(S₁)≥length(S₂), so the shape is [d,0] for some d≥0 called the defect of the symbol; the content is equal to mod(d,2). For symbols [S₁,S₂] with length(S₁)==length(S₂) we choose representatives such that P₁≤P₂ lexicographically where P₁,P₂ are the partitions associated to S₁,S₂.

Partitions and pairs of partitions parametrize characters of the Weyl groups of classical types, and tuples of partitions parametrize characters of imprimitive complex reflection groups. 2-Symbols parametrize unipotent characters of classical Chevalley groups, and more general e-symbols parametrize unipotent characters of spetses associated to spetsial imprimitive complex reflection groups. The rank of a symbol is equal to the semi-simple rank of the corresponding Chevalley group or Spets.

Symbols of rank n and defect 0 parametrize characters of the Weyl group W(Dₙ), and symbols of rank n and defect≡0 (mod 4) parameterize unipotent characters of SO₂ₙ; symbols of rank n and defect≡2 (mod 4) parameterize unipotent characters of SO⁻₂ₙ. Symbols of rank n and defect 1 parametrize characters of the Weyl group W(Bₙ), and symbols of rank n and odd defect parametrize unipotent characters of Sp₂ₙ or SO₂ₙ₊₁.

To parametrize unipotent characters of spetses we need another statistic on symbols, the Malle-defect defined by Malle as mod(binomial(e,2)*div(sum(length.(S)),e)-dot(O:e-1,length.(S)),e). It is invariant under shift; it is invariant under circular permutation for symbols of content O but not for symbols of content 1; for these only one circular permutation has Malle-defect 0.

e-symbols of rank n and content 1 parameterize unipotent characters of the spets G(e,1,n). The principal series (in bijection with characters of the reflection group) is parametrized by symbols such that tje reduced ones have shape [1,0,…,0].

Unipotent characters of the spets G(e,e,n) are parametrized by e-symbols of content 0 and Malle-defect 0. The symbols for the principal series characters have shape [0,…,0].

Unipotent characters of the twisted spets ᵗG(e,e,n) are parametrized by e-symbols of rank n, content 0 and Malle-defect t.

In the above parametrizations, periodic symbols, that is symbols S such that the sequence S₁,…,Sₙ is a repetition k times of a shorter sequence, must be repeated k times. To distinguish the k copies, an additional parameter, a k-th root of unity, is attached to the symbol. As an example, here are the symbols for G₃,₃,₃:

julia> symbols(3,3,0)
12-element Vector{CharSymbol}:
 (1+)
 (1ζ₃)
 (1ζ₃²)
 (01,12,02)
 (01,02,12)
 (012,012,123)
 (0,1,2)
 (0,2,1)
 (01,01,13)
 (0,0,3)
 (012,,)
 (012,012,)

when the symbol has a period, only the period is shown, followed by the root of unity (where 1 is shown as + and -1 is shown as -.

The functions for symbols in this module are

CharSymbol, which constructs a symbol
Symbol_partition_tuple which constructs a symbol of a given shape from a partition tuple
rank which computes the rank of a symbol
defect which returns the defect of a 2-symbol
Malledefect which returns the Malle-defect of a symbol
fakedegree, degree_feg, valuation_feg return the fake degree (resp. its degree and valuation) of the unipotent character parametrized by a symbol
gendeg, degree_gendeg, valuation_gendeg return the generic degree (resp. its degree and valuation) of the unipotent character parametrized by a symbol
symbols returns the list of symbols of a given length, rank and content.

Finally, in this module we also provides a function XSP which returns the "symbols" (pairs of lists of increasing positive integers satisfying some conditions) $X̃^{ρ-s,s}_{n,d}$ defined by Lusztig and Spaltenstein which parametrize local systems on unipotent classes for classical reductive groups.