Amazing package providing symbolic but explicit

• decomposition of group representations into (semi-)simple representations and
• the Wedderburn decompositions for endomorphisms of those representations.

We work with

• a (linear) unitary actions of a finite group G on finite dimensional vector space V over K = ℝ or K = ℂ (i.e. a KG-module V) and
• linear, G-equivariant maps (equivariant endomorphisms) f : V → V.

These objects are of primary inportance in the study of the representation theory for finite groups, but also naturally arise from (non)commutative polynomial optimization with group symmetry. The aim of the package is to facilitate such uses.

By Maschke's theorem V can be decomposed uniquely V ≅ V₁ ⊕ ⋯ ⊕ Vᵣ into isotypic/semisimple subspaces Vᵢ and each of Vᵢ ≅ mᵢWᵢ is (in a non-canonical fashion) isomorphic to a direct sum of mᵢ copies of irreducible/simple subspaces Wᵢ. By (symbolic) computation in (the group algebra) KG, SymbolicWedderburn is capable of producing the exact isomorphism V ≅ V₁ ⊕ ⋯ ⊕ Vᵣ in the form of a collection of projections πᵢ : V → Vᵢ either in the group algebra (lazy, unevaluated form), or in terms of projection matrices, when a basis for V is explicitly given.

The isomorphism produces a decomposition of End_G(V) (the set of linear G-equivariant self-maps of V) in the sense of Artin-Wedderburn theorem, i.e. the projections πᵢ block-diagonalize f ≅ f₁ ⊕ ⋯ ⊕ fᵣ where fᵢ : Vᵢ → Vᵢ. In terms of matrices if f is given by n×n-matrix, then we can rewrite it as a block diagonal matrix with blocks of sizes nᵢ×nᵢ where Σᵢ nᵢ = n = dim V and each nᵢ = mᵢ · dim Wᵢ.

For example, if a basis for a semi-definite constraint admits an action of a finite group, then the semi-definite matrix of a invariant solution is such an equivariant endomorphism. In particular if P is a positive semidefinite constraint, when searching for an G-invariant solution we may replace

0 ⪯ P[1:n, 1:n]

by a sequence of constraints

0 ⪯ P[1:nᵢ, 1:nᵢ] for i = 1…r,

greatly reducing the computational complexity: the size of the psd constraint is reduced from n² to Σᵢ nᵢ². Such replacement can be justified if e.g. the objective is symmetric and the set of linear constraints follows a similar group-symmetric structure.

If we are only interested in the feasibility of an optimization problem, then such replacement is always justified (i.e. an invariant solution is a honest solution which might not attain the same objective).

Sometimes even stronger reduction is possible when the acting group G is sufficiently complicated and we have a minimal projection system at our disposal (The package tries to compute such system by a heuristic algorithm. If the approach fails please open an issue!). In such case we can (often) find subsequent projections Vᵢ → K^{mᵢ} (depending only on the multiplicity of the irreducible, not on its dimension!). This leads to an equivalent formulation for the psd constraint with nᵢ = mᵢ further reducing its size.

Moreover in the case of symmetric optimization problems it's possible to use the symmetry to reduce the number of linear constraints (since in that case only one constraint per orbit is needed). SymbolicWedderburn facilitates also this simplification.

In Aut(𝔽₅) has property (T) we use the trick above to successfully simplify and solve a large semidefinite problem coming from sum-of-squares optimization.

The original problem had one (symmetric) psd constraint of size 4641×4641 and 11_154_301 linear constraints. By exploiting its (admittedly -- pretty large) symmetry group (of order 3840) we can reduce this problem to 20 (symmetric) psd constraints of sizes

[56  38  34  32  27  27  23  23  22  22  18  17  9  8  6  2  1  1  1  1]

which correspond to (the simple) Wᵢ blocks above. In particular, the number of variables in psd constraints was reduced from 10_771_761 to just 5_707.

Moreover, the symmetry group has just 7 229 orbits (when acting on the subspace of linear constraints), so the symmetrized problem has equal number of (a bit denser) linear constraints.

The symmetrized problem is solvable in ~20 minutes on an average office laptop (with 16GB of RAM).

For more examples you may have a look at dihedral action example, or different sum of squares formulations.

This package is used by SumOfSquares to perform exactly this reduction, for an example use see its documentation.

The software for sum of (hermitian) squares computations in a non-commutative setting (group algebra of a infinite group) using SymbolicWedderburn is my project PropertyT.jl (unregistered). There we used the sum of squares optimization to prove Property (T) for special automorphisms group of the free group. It's a cool result, check it out!.

The main aim of GAP package Wedderga is to

compute the simple components of the Wedderburn decomposition of semisimple group algebras of finite groups over finite fields and over subfields of finite cyclotomic extensions of the rationals.

The focus is thus on symbolic computations and identifying isomorphism type of the simple components. SymbolicWedderburn makes no efforts to compute the types or defining fields, it's primary goal is to compute symbolic/numerical Wedderburn-Artin isomorphism in a form usable for (polynomial) optimization. Wedderga also contains much more sophisticated methods for computing a complete set of orthogonal primitive idempotents (i.e. a minimal projection system) through Shoda pairs. In principle those idempotents could be computed using Oscar and used in SymbolicWedderburn.

If you happen to use SymbolicWedderburn please cite either of

• M. Kaluba, P.W. Nowak and N. Ozawa $Aut(F₅)$ has property (T) 1712.07167, and
• M. Kaluba, D. Kielak and P.W. Nowak On property (T) for $Aut(Fₙ)$ and $SLₙ(Z)$ 1812.03456.

(Follow the arxiv link for proper link to the journal.)

The algorithm of SymbolicWedderburn.jl can be summarised in a few steps. As an example it might be helpful to think of G acting on a polynomial ring by permuting variables.

1. Given the action of G on variables the action η on the whole monomial basis is induced and therefore on the whole polynomial ring.
2. For the concrete G-invariant linear subspace V (given by a fixed set of monomials) of the polynomial ring (a vector space with G-action) we compute the character of representation η:G → 𝒰(V) (we assume that η is unitary, which is automatic for (signed) permutation actions).

3. After computing the character table of G we find the decomposition of (V, η) into irreducibles: η ≅ χ₁ + … + χᵣ by some symbolic magic (i.e. orthogonality relations) in the group ring ℂG (or ℝG). We know that this decomposition corresponds to a decomposition V ≅ V₁ ⊕ ⋯ ⊕ Vᵣ, and we will compute that correspondence explicitely in a moment.
4. Moreover, we abstractly know that each χᵢ ≅ mᵢϱᵢ (and Vᵢ ≅ mᵢWᵢ) is isomorphic to a multiple mᵢ of irreducible characters ϱᵢ, but we can't use this information yet. These irreducibles lead us later to πᵢ : V → Vᵢ, the projections onto isotypical summands.
   show me more…

   Our implementation of projections is matrix-free. Projections are just idempotent elements (x² = x) in the group algebra. In that sense the projection to an isotypical component is unique in ℂG, but not as πᵢ = η(ϱᵢ), an element of End_G((V, η)) i.e. as a matrix: a matrix representation of a projection already includes a choice of basis (think of the difference of a linear operator vs its matix).

4. Sometimes this step is followed by finding even tighter minimal projection system

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   Finding tighter projections use a lemma of Schur.

   Lemma (Schur) Over an algebraically closed field the commutant of a matrix algebra consist of matrices of a particularly simple form:

   • direct sums of endomorphisms of isotypical subspaces (i.e. isotypical subspaces are orthogonal which gives us block structure for endomorphisms),
   • within isotypical subspace (of character ϱ) the endomorphisms are of the form M⊗Iₙ, where n = degree(ϱ) and M is (square) of size m = multiplicity(ϱ, η).

   Here the matrix algebra is the one defined by the image of η and the projections commute with those, so the conclusion is that reconstructing a single projection endomorphism corresponding to an irreducible ϱ requires only m² parameters (regardless of the degree of ϱ!)

   • For every irreducible character ϱᵢ we try to find a (non-central) projection pᵢ such that ϱᵢ∘pᵢ(e) = k is as small as possible (desirably just 1), so that η(ϱᵢ∘pᵢ) = πᵢ∘η(pᵢ) and therefore rank(πᵢ∘η(pᵢ)) = k. We call those {ϱᵢ∘pᵢ}_{ϱ ∈ Irr(G)} a minimal projection system. Note: the existence and complexity of finding the system depends on the group only, not on the representation η, its associated action (and hence not on the dimension of V!).
   • Sometimes the system exists (symmetric, alternating groups etc.) sometimes it doesn’t (e.g. real representations of cyclic groups).
   • We employ a simple brute-force algorithm to search for pᵢ over all characteristic/alternating projections for small subgroups of G.

5. Given the monomial basis we realize those projections as sparse matrices (only now we start computing with matrices, but even this step is exact).
6. Unitary vectors in the images of those projections are found via sparse qr decomposition and these basis vectors form the symmetry adapted basis.
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   The image (i.e. as a linear subspace) of the matrix projection is well defined; any (orthogonal) basis of the subspace would do; We just take the first few columns of the Q factor of sparse qr factorisation.

For more complete introduction to projections, characters and their place in the group ring we recommend the book by J.P. Serre Linear representations of finite groups. A somewhat condensed account of minimal projection system is presented in sections 2 (theory) and 3.3 (particular example computations) of 1712.07167.