Unnamed group, Coefficient()

Author: joelgibson

MuCoeffs/MuCoeffs.m

@@ -67,7 +67,7 @@ intrinsic _MuCoeffFromTable(table::Rec, w::GrpFPCoxElt) -> Assoc
 
 intrinsic _LoadMuTable(W::GrpFPCox) -> BoolElt, Assoc
 {Load the mu-table for a Coxeter group, returning (true, table) if one exists, and (false, false) otherwise.}
-    path := mucoeff_dir() cat "/" cat CartanName(W) cat ".mus";
+    path := mucoeff_dir() cat "/" cat _IHkeCartanName(W) cat ".mus";
     ok, io := OpenTest(path, "r");
     if not ok then
         return false, false;

Package/Cells.m

@@ -75,7 +75,7 @@ intrinsic EnumerateCoxeterGroup(W::GrpFPCox: lengthBound := -1, quiet := false)
 intrinsic EnumerateCoxeterGroup(W::GrpFPCox, I::SeqEnum[RngIntElt]: lengthBound:=-1, quiet:=false) -> SetIndx[GrpFPCoxElt]
 {The same as EnumerateCoxeterGroup, but enumerating only the I-minimal elements.}
     warningLimit := 1000000;
-    error if lengthBound lt 0 and not IsCoxeterFinite(CartanName(W)),
+    error if lengthBound lt 0 and not IsCoxeterFinite(CoxeterMatrix(W)),
         "Cannot enumerate an infinite group.";
     lengthBound := lengthBound lt 0
         select #LongestElement(W)

Package/EltIHke.m

@@ -66,14 +66,23 @@ intrinsic FreeModule(elt::EltIHke) -> FModIHke
 end intrinsic;
 
 intrinsic Coefficient(elt::EltIHke, w::GrpFPCoxElt) -> RngElt
-{Return the coefficient of w in the linear combination.}
+{Return the coefficient of B(w) in the linear combination, where B is the basis of elt.}
     if IsDefined(elt`Terms, w) then
         return elt`Terms[w];
     else
         return BaseRing(Parent(elt)) ! 0;
     end if;
 end intrinsic;
 
+intrinsic Coefficient(elt::EltIHke, w::GrpFPCoxElt, d::RngIntElt) -> RngIntElt
+{Return the coefficient of B(w) v^d in the linear combination, where B is the basis of elt.}
+    if IsDefined(elt`Terms, w) then
+        return Coefficient(elt`Terms[w], d);
+    else
+        return 0;
+    end if;
+end intrinsic;
+
 intrinsic Support(elt::EltIHke) -> SetIndx[GrpFPCoxElt], SeqEnum[RngElt]
 {Return two parallel sequences giving the group elements and coefficients in the linear combination.
  The support (the group elements) are returned as an indexed set, which can be converted to a

Package/FModIHke.m

@@ -5,6 +5,15 @@ intrinsic IHeckeVersion() -> MonStgElt
     return "IHecke version 2021-11-01";
 end intrinsic;
 
+intrinsic _IHkeCartanName(W::GrpFPCox) -> MonStgElt
+{Return the Cartan name of W if it is finite or affine, and otherwise something like X9.}
+    try
+        return CartanName(W);
+    catch e
+        return Sprintf("X%o", Rank(W));
+    end try;
+end intrinsic;
+
 
 ///////////////////////////////
 // Free module (abstract class)

Package/IHkeAlg.m

@@ -8,7 +8,7 @@ intrinsic IHeckeAlgebra(W::GrpFPCox) -> IHkeAlg
     // Magma has no Laurent polynomials, but we can use the polynomial part of the series ring.
     LPolyRing<v> := LaurentSeriesRing(Integers());
     HAlg := New(IHkeAlg);
-    name := Sprintf("Iwahori-Hecke algebra of type %o", CartanName(W));
+    name := Sprintf("Iwahori-Hecke algebra of type %o", _IHkeCartanName(W));
     _FModIHkeInit(~HAlg, LPolyRing, name, W);
     return HAlg;
 end intrinsic;
@@ -34,7 +34,7 @@ intrinsic IHeckeASMod(W::GrpFPCox, I::SeqEnum[RngIntElt]) -> IHkeASMod
     LPolyRing<v> := LaurentSeriesRing(Integers());
 
     asmod := New(IHkeASMod);
-    name := Sprintf("Antispherical module of type %o, parabolic %o", CartanName(W), I);
+    name := Sprintf("Antispherical module of type %o, parabolic %o", _IHkeCartanName(W), I);
     _FModIHkeInit(~asmod, LPolyRing, name, W);
     asmod`Para := I;
     return asmod;
@@ -66,7 +66,7 @@ intrinsic IHeckeSMod(W::GrpFPCox, I::SeqEnum[RngIntElt]) -> IHkeSMod
     LPolyRing<v> := LaurentSeriesRing(Integers());
 
     smod := New(IHkeSMod);
-    name := Sprintf("Spherical module of type %o, parabolic %o", CartanName(W), I);
+    name := Sprintf("Spherical module of type %o, parabolic %o", _IHkeCartanName(W), I);
     _FModIHkeInit(~smod, LPolyRing, name, W);
     smod`Para := I;
     return smod;
@@ -80,4 +80,4 @@ intrinsic IHeckeSMod(W::GrpFPCox, I::SeqEnum[RngIntElt]) -> IHkeSMod
 intrinsic Parabolic(SMod::IHkeSMod) -> SeqEnum[RngIntElt]
 {The parabolic generators.}
     return SMod`Para;
-end intrinsic;
+end intrinsic;

Package/IHkeAlgCan.m

@@ -171,7 +171,7 @@ intrinsic CanonicalBasis(HAlg::IHkeAlg : UseTable := true) -> IHkeAlgCan
         ok, table := _LoadMuTable(CoxeterGroup(HAlg));
         if ok then
             basis`MuTable := table;
-            vprintf IHecke: "IHecke: Mu coefficients for %o loaded from the database\n", CartanName(CoxeterGroup(HAlg));
+            vprintf IHecke: "IHecke: Mu coefficients for %o loaded from the database\n", _IHkeCartanName(CoxeterGroup(HAlg));
         end if;
     end if;
     return basis;

README.md

@@ -169,7 +169,7 @@ by checking the version:
     $ magma
     > AttachSpec("IHecke.spec");
     > IHeckeVersion();
-    IHecke version 2021-09-10
+    IHecke version 2021-11-01
 
 Now, create a `GrpFPCox` of your favourite type:
 
@@ -353,6 +353,17 @@ In order to extract a particular coefficient, use the `Coefficient` function.
     > Coefficient(H.[2,1] * H.1, W.1);
     0
 
+The `Coefficient` function can be given an extra argument to extract the v^d coefficient from the Laurent polynomial.
+
+    > Coefficient(H.[2,1] * H.1, W![2,1]);
+    v^-1 - v
+    > Coefficient(H.[2,1] * H.1, W![2,1], 1);
+    -1
+    > Coefficient(H.[2,1] * H.1, W![2,1], 0);
+    0
+    > Coefficient(H.[2,1] * H.1, W![2,1], -1);
+    1
+
 The Kazhdan-Lusztig bar involution can be calculated using `Bar`.
 
     > Bar(H.1);
@@ -762,6 +773,8 @@ We aim to keep these to a minimum once the package is in use.
 - Development version
   - Added a faster Standard x Canonical -> Canonical multiplication.
   - The cell order relations are precomputed, making cell order testing much faster.
+  - Fixed a crash when creating a Hecke algebra for a group not of affine or finite type.
+  - Added a third argument to `Coefficient()` for extracting the v^d term.
 - Version 2021-11-01
   - Added an experimental "literal" basis type (a basis specified by a partial table). I will wait
       to see how it plays out in other projects before making it a feature.
@@ -789,6 +802,9 @@ We aim to keep these to a minimum once the package is in use.
 
 # TODO
 
+- (High priority) Enhance the Literal basis
+  - Allow it to be a partial basis (eg for affine groups).
+  - Make some standard way to save and restore it, probably through MAGMA object files.
 - (High priority) Allow the multiplication `C(w) * C.s` to be driven by a table of mu-coefficients
     (in other words, a W-graph) so that this special case is extremely fast. By induction (and the
     standard multiplication formula), this means that multiplication within the canonical basis can

Tests/TestBase.m

@@ -37,6 +37,11 @@
 assert Coefficient(H.0, W.0) eq 1;
 assert Coefficient(H.0, W.1) eq 0;
 
+// Coefficient for B(w) v^d.
+assert Coefficient(H.0, W.0, 0) eq 1;
+assert Coefficient(H.0, W.0, 1) eq 0;
+assert Coefficient(H.0 -3 * v^2 * H.1, W.1, 2) eq -3;
+
 // Standard basis coercion from scalars.
 assert IsZero(H ! 0);
 assert not IsZero(H ! 1);

Tests/TestUnnamedGroup.m

@@ -0,0 +1,22 @@
+// Test creation of the Hecke algebra for a non-standard group.
+
+SetQuitOnError(true);
+SetColumns(0);
+SetAssertions(3);
+AttachSpec("IHecke.spec");
+
+// The Coxeter graph for a triangle with all sides labelled by 5.
+W := CoxeterGroup(GrpFPCox, Matrix(Integers(), [
+    [1, 5, 5],
+    [5, 1, 5],
+    [5, 5, 1]
+]));
+HAlg := IHeckeAlgebra(W);
+assert Sprint(HAlg) eq "Iwahori-Hecke algebra of type X3";
+
+H := StandardBasis(HAlg);
+C := CanonicalBasis(HAlg);
+x := H ! C.[1,2,3,2,1];
+
+print "TestUnnamedGroup passed.";
+quit;

lint.py

@@ -15,7 +15,9 @@
 ]
 
 def check_file(fname: str):
-    for line_no, line in enumerate(open(fname), start=1):
+    lines = list(open(fname))
+    for i, line in enumerate(lines):
+        line_no = i + 1
         line = line.rstrip('\n')
 
         if '\t' in line:
@@ -30,6 +32,10 @@ def check_file(fname: str):
         if 'todo' in line.lower():
             print(f"{fname}:{line_no} TODO found: {line}")
 
+        # Look for CartanName() being called, not preceded by a try statement.
+        if re.search(r'(?<!_IHke)CartanName\(', line) and not (i > 0 and 'try' in lines[i-1]):
+            print(f"{fname}:{line_no} calls CartanName(), it should call _IHkeCartanName()")
+
 
 for fname in SOURCE_FILES:
     check_file(fname)