From 7e18eb5ffa0b6d95a3ee9612627c7d4102dfb368 Mon Sep 17 00:00:00 2001
From: Khashayar Neshat <111620549+kneshat@users.noreply.github.com>
Date: Tue, 9 Jul 2024 06:50:33 -0700
Subject: [PATCH 1/4] Add files via upload

Adding an example to illustrate how real irreps are constructed from complex ones in the three different cases of FS=1, 0, and -1.
---
 examples/ex_SL(2,3).jl | 35 +++++++++++++++++++++++++++++++++++
 1 file changed, 35 insertions(+)
 create mode 100644 examples/ex_SL(2,3).jl

diff --git a/examples/ex_SL(2,3).jl b/examples/ex_SL(2,3).jl
new file mode 100644
index 0000000..efed238
--- /dev/null
+++ b/examples/ex_SL(2,3).jl
@@ -0,0 +1,35 @@
+using SymbolicWedderburn
+using PermutationGroups
+import SymbolicWedderburn as SW
+
+
+# Constructing SL(2,3) or binary tetrahedral group as a permutation group of 8 elements
+gen1 = perm"(1,2,3,4)(5,6,7,8)" 
+gen2 = perm"(1,7,3,5)(2,6,4,8)"
+gen3 = perm"(2,6,7)(4,8,5)"  
+MyGroup = PermGroup([gen1, gen2, gen3]) # SL(2,3)
+
+# Compute the character table
+tbl = SW.CharacterTable(Rational{Int}, MyGroup)
+
+# Get irreducible characters
+irreducible_chars = irreducible_characters(tbl)
+
+# Define multiplicities (for simplicity, use ones)
+multiplicities = fill(1, length(irreducible_chars))
+
+# Get real irreducible characters and their multiplicities
+real_irreps, real_mults = SW.affordable_real(irreducible_chars, multiplicities)
+# Print the Frobinus-Schur indicator of each complex irrep
+using SymbolicWedderburn.Characters
+for (i, χ) in enumerate(irreducible_characters(tbl))
+    fs_indicator = Characters.frobenius_schur(χ)
+    println("Frobenius-Schur indicator of $χ: $fs_indicator")
+    println()
+end
+# Print characters of real irreps
+println("Real Irreducible Characters:")
+for irrep in real_irreps
+    println(irrep)
+end
+# Expected Result: χ does not change if its FS is 1. χ is doubled if its FS is -1. χ is added by another character if its FS is 0.
\ No newline at end of file

From dafd813001368b973d9190ff87f6645d8c8b4b08 Mon Sep 17 00:00:00 2001
From: Khashayar Neshat <111620549+kneshat@users.noreply.github.com>
Date: Tue, 9 Jul 2024 14:27:18 -0700
Subject: [PATCH 2/4] Update ex_SL(2,3).jl

---
 examples/ex_SL(2,3).jl | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/examples/ex_SL(2,3).jl b/examples/ex_SL(2,3).jl
index efed238..3cc15b3 100644
--- a/examples/ex_SL(2,3).jl
+++ b/examples/ex_SL(2,3).jl
@@ -13,7 +13,7 @@ MyGroup = PermGroup([gen1, gen2, gen3]) # SL(2,3)
 tbl = SW.CharacterTable(Rational{Int}, MyGroup)
 
 # Get irreducible characters
-irreducible_chars = irreducible_characters(tbl)
+irreducible_chars = SW.irreducible_characters(tbl)
 
 # Define multiplicities (for simplicity, use ones)
 multiplicities = fill(1, length(irreducible_chars))
@@ -32,4 +32,4 @@ println("Real Irreducible Characters:")
 for irrep in real_irreps
     println(irrep)
 end
-# Expected Result: χ does not change if its FS is 1. χ is doubled if its FS is -1. χ is added by another character if its FS is 0.
\ No newline at end of file
+# Expected Result: χ does not change if its FS is 1. χ is doubled if its FS is -1. χ is added by another character if its FS is 0.

From 0161efd7e5441a25bcf068f582c4dc9a7f79db98 Mon Sep 17 00:00:00 2001
From: Khashayar Neshat <111620549+kneshat@users.noreply.github.com>
Date: Tue, 9 Jul 2024 19:59:22 -0700
Subject: [PATCH 3/4] Update sa_basis.jl

Distinguishing the construction of real-type and quaternion-type real characters.
---
 src/sa_basis.jl | 8 ++++++--
 1 file changed, 6 insertions(+), 2 deletions(-)

diff --git a/src/sa_basis.jl b/src/sa_basis.jl
index 0403425..cc8b315 100644
--- a/src/sa_basis.jl
+++ b/src/sa_basis.jl
@@ -6,10 +6,14 @@ function affordable_real(
     mls_real = similar(multiplicities, 0)
     for (i, χ) in pairs(irreducible_characters)
         ι = Characters.frobenius_schur(χ)
-        if abs(ι) == 1 # real or quaternionic
-            @debug "real/quaternionic:" χ
+        if ι == 1 # real
+            @debug "real" χ
             push!(irr_real, χ)
             push!(mls_real, multiplicities[i])
+        elseif ι == -1 # quaternion
+            @debug "quaterionic" χ
+            push!(irr_real, 2*χ)
+            push!(mls_real, multiplicities[i]/2)
         else # complex one...
             cχ = conj(χ)
             k = findfirst(==(cχ), irreducible_characters)

From 5b257318909a85ae93751e5c1ad96b0e510ad650 Mon Sep 17 00:00:00 2001
From: Khashayar Neshat <111620549+kneshat@users.noreply.github.com>
Date: Tue, 9 Jul 2024 20:09:03 -0700
Subject: [PATCH 4/4] Update ex_SL(2,3).jl

---
 examples/ex_SL(2,3).jl | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/examples/ex_SL(2,3).jl b/examples/ex_SL(2,3).jl
index 3cc15b3..1fd7144 100644
--- a/examples/ex_SL(2,3).jl
+++ b/examples/ex_SL(2,3).jl
@@ -15,8 +15,8 @@ tbl = SW.CharacterTable(Rational{Int}, MyGroup)
 # Get irreducible characters
 irreducible_chars = SW.irreducible_characters(tbl)
 
-# Define multiplicities (for simplicity, use ones)
-multiplicities = fill(1, length(irreducible_chars))
+# Define multiplicities (for simplicity, use twos, it should be even for quaternion-type irreps)
+multiplicities = fill(2, length(irreducible_chars))
 
 # Get real irreducible characters and their multiplicities
 real_irreps, real_mults = SW.affordable_real(irreducible_chars, multiplicities)