From e3290a5c4686b66b415fd5e0ff28b129d5f5b30e Mon Sep 17 00:00:00 2001
From: Marek Kaluba <kalmar@amu.edu.pl>
Date: Sun, 12 Sep 2021 12:51:11 +0200
Subject: [PATCH] =?UTF-8?q?minimal=20rank=20projs:=20add=20support=20for?=
 =?UTF-8?q?=20=CF=87=20+=20conj(=CF=87)?=
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---
 src/minimal_projections.jl | 18 +++++++++++++-----
 src/sa_basis.jl            | 13 +++++++++----
 test/action_permutation.jl |  6 +++---
 test/sa_basis.jl           |  5 +++--
 4 files changed, 28 insertions(+), 14 deletions(-)

diff --git a/src/minimal_projections.jl b/src/minimal_projections.jl
index 5f65a87..c2c6c84 100644
--- a/src/minimal_projections.jl
+++ b/src/minimal_projections.jl
@@ -95,20 +95,28 @@ else
     end
 end
 
-function rank_one_projection(χ::Character, RG::StarAlgebra{<:Group};
+function minimal_rank_projection(χ::Character, RG::StarAlgebra{<:Group};
     idems = small_idempotents(RG), # idems are AlgebraElements over Rational{Int}
     iters=3
 )
-    degree(χ) == 1 && return one(RG, Rational{Int}), true
+
+    k = if isirreducible(χ)
+        1
+    else
+        sum(constituents(χ).>0)
+        # the minimal rank projection for the composite character
+    end
+
+    degree(χ) == k && return one(RG, Rational{Int}), true
 
     for µ in idems
-        isone(χ(µ)) && µ^2 == µ && return µ, true
+        χ(µ) == k && µ^2 == µ && return µ, true
     end
 
     for n in 2:iters
         for elts in Iterators.product(ntuple(i -> idems, n)...)
             µ = *(elts...)
-            isone(χ(µ)) && µ^2 == µ && return µ, true
+            χ(µ) == k && µ^2 == µ && return µ, true
         end
     end
     @debug "Could not find minimal projection for $χ"
@@ -119,7 +127,7 @@ function minimal_projection_system(
     chars::AbstractVector{<:AbstractClassFunction},
     RG::StarAlgebra{<:Group}
 )
-    res = fetch.([@spawn_compat rank_one_projection(χ, RG) for χ in chars])
+    res = fetch.([@spawn_compat minimal_rank_projection(χ, RG) for χ in chars])
 
     r1p, simple = first.(res), last.(res) # rp1 are sparse storage
 
diff --git a/src/sa_basis.jl b/src/sa_basis.jl
index 452565d..0ddb75a 100644
--- a/src/sa_basis.jl
+++ b/src/sa_basis.jl
@@ -217,11 +217,16 @@ function _symmetry_adapted_basis(
         @spawn_compat begin
             µT = eltype(µ) == T ? µ : AlgebraElement{T}(µ)
             image = hom === nothing ? image_basis(µT) : image_basis(hom, µT)
-            if simple
-                @assert size(image, 1) == m "The dimension of the projection doesn't match with simple summand multiplicity"
-            end
             DirectSummand(image, m, deg, simple)
         end
     end
-    return fetch.(res)
+    direct_summands = fetch.(res)
+
+    for (χ, ds) in zip(irr, direct_summands)
+        if issimple(ds) && (d = size(ds, 1)) != (e = multiplicity(ds)*sum(constituents(χ).>0))
+            throw("The dimension of the projection doesn't match with simple summand multiplicity: $d ≠ $e")
+        end
+    end
+
+    return direct_summands
 end
diff --git a/test/action_permutation.jl b/test/action_permutation.jl
index c0c46c1..cc1495f 100644
--- a/test/action_permutation.jl
+++ b/test/action_permutation.jl
@@ -96,7 +96,7 @@ end
         end
 
         let χ = irr[1], m = multips[1]
-            (a, fl) = SymbolicWedderburn.rank_one_projection(χ, RG)
+            (a, fl) = SymbolicWedderburn.minimal_rank_projection(χ, RG)
             @test fl == isone(χ(a))
 
             µ = AlgebraElement(χ, RG)*a
@@ -116,7 +116,7 @@ end
         end
 
         let χ = irr[2], m = multips[2]
-            (a, fl) = SymbolicWedderburn.rank_one_projection(χ, RG)
+            (a, fl) = SymbolicWedderburn.minimal_rank_projection(χ, RG)
             @test fl == isone(χ(a))
 
             µ = AlgebraElement(χ, RG)*a
@@ -126,7 +126,7 @@ end
         end
 
         let χ = irr[3], m = multips[3]
-            (a, fl) = SymbolicWedderburn.rank_one_projection(χ, RG)
+            (a, fl) = SymbolicWedderburn.minimal_rank_projection(χ, RG)
             @test fl == isone(χ(a))
 
             µ = AlgebraElement(χ, RG)*a
diff --git a/test/sa_basis.jl b/test/sa_basis.jl
index d6a09a0..807450d 100644
--- a/test/sa_basis.jl
+++ b/test/sa_basis.jl
@@ -34,7 +34,7 @@ end
             StarAlgebra(G, b, (length(b), length(b)))
         end
 
-        µ, s = SymbolicWedderburn.rank_one_projection(irr[1], RG)
+        µ, s = SymbolicWedderburn.minimal_rank_projection(irr[1], RG)
         @test µ isa AlgebraElement
         @test s
 
@@ -129,7 +129,8 @@ end
                     sa_basisR = symmetry_adapted_basis(Float64, G, S, semisimple=false)
                     for b in sa_basisR
                         if issimple(b)
-                            @test multiplicity(b) == size(b, 1)
+                            @test multiplicity(b) == size(b, 1) || 2*multiplicity(b) == size(b, 1)
+                            # the first condiditon doesn't hold for realified characters;
                         else
                             @test multiplicity(b)*SymbolicWedderburn.degree(b) == size(b, 1)
                         end