WIP: Classical groups. Express elements in LGO (nice) generators.

gap/matrix/wordsInNiceGens/BruhatDecomposition.gd

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+#############################################################################
+# BruhatDecomposition.gd
+#############################################################################
+#############################################################################
+##
+##  BruhatDecomposition package
+##
+##    Daniel Rademacher, RWTH Aachen University
+##    Alice Niemeyer, RWTH Aachen University
+##
+## Licensed under the GPL 3 or later.
+##
+#############################################################################
+#
+#! @Chapter Foreword
+#!
+#! Let <M>G</M> be one of the classical groups SL, Sp, SU or SO over a finite field of size <M>q</M> and dimension <M>d</M>. Let g be an element in G.
+#! We want to write <M>g = u_1 \cdot w \cdot u_2</M> with <M>u_1</M> and <M>u_2</M> lower unitriangular matrices and <M>w</M> a monomial matrix. <P/>
+#! This is already implemented for:
+#! <List>
+#!     <Item>
+#!     Special linear group (SL) (see Chapter <Ref Sect="Chapter_SpecialLinearGroup"/>)
+#!     </Item>
+#!     <Item>
+#!     Symplectic group (Sp) (see Chapter <Ref Sect="Chapter_SymplecticGroup"/>)
+#!     </Item>
+#!     <Item>
+#!     Special unitary group (SU) (see Chapter <Ref Sect="Chapter_SpecialUnitaryGroup"/>)
+#!     </Item>
+#!     <Item>
+#!     Special orthogonal group (SO) (see Chapter <Ref Sect="Chapter_SpecialOrthogonalGroup"/>)
+#!     </Item>
+#! </List>
+
+
+
+#! @Section Main Function
+#! @SectionLabel MainFunction
+
+#####
+# BruhatDecomposition()
+#####
+
+#! @Arguments g
+#! @Returns pgr (A SLP to compute <M>u_1,u_2,p_{sign}</M> and <M>diag</M> and the matrices <M>u_1, u_2, p_{sign}</M> and <M>diag</M> itself.)
+#! @Description
+#! Checks whether <M>g</M> is an element of one of the classical groups in their natural representation. If yes, the corresponding Bruhat decomposition of the group and the element <M>g</M> is calculated. Otherwise the function prints a warning.
+DeclareGlobalFunction( "BruhatDecomposition" );

gap/matrix/wordsInNiceGens/BruhatDecomposition.gi

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+######################################
+# BruhatDecomposition.gi
+######################################
+
+#####
+# BruhatDecomposition()
+#####
+
+InstallGlobalFunction(  BruhatDecomposition,
+function(g)
+
+    local d, fld, q;
+
+    d := Length( g );
+    fld := FieldOfMatrixList( [g] );
+    q := Size(fld);
+
+    if g in Sp(d,q) then
+        return BruhatDecompositionSp(LGOStandardGensSp(d,q),g);
+    elif g in SU(d,q) then
+        return BruhatDecompositionSU(LGOStandardGensSU(d,q),g);
+    elif g in SO(1,d,q) then
+        return BruhatDecompositionSO(LGOStandardGensSO(1,d,q),g);
+    elif g in SO(0,d,q) then
+        return BruhatDecompositionSO(LGOStandardGensSO(0,d,q),g);
+    elif g in SO(-1,d,q) then
+        return BruhatDecompositionSO(LGOStandardGensSO(-1,d,q),g);
+    elif g in SL(d,q) then
+        return BruhatDecompositionSL(LGOStandardGensSL(d,q),g);
+    else
+        Print("The element g is not an element of one of the classical groups in their natural representation. \n");
+        Print("Abort.");
+    fi;
+
+end
+);