# # SL2Reps: Constructing symmetric representations of SL(2,Z). # # Lists of Representations. # # Implementations #
InstallGlobalFunction( _SL2IrrepsPPLOfDegree, function(degree)
local irrep_list, p, ld, pmax, ldmax, pset, ldset, name, rho, l,
i, j, u, w, si, r, t, x;
if not degree in PositiveIntegers then
Error("degree must be a positive integer."); fi;
irrep_list := [];
if1 = degree then # The trivial irrep Xi_0 = C_1; level 1.
name := "Xi_0";
_SL2RecordIrrep(irrep_list, name, [[[1]], [[1]]], 1); fi;
# p odd, ld = 1.
pmax := 2*degree + 1;
pset := Filtered([3..pmax], x -> IsPrime(x)); for p in pset do
l := p^1; if p + 1 = degree and p > 3then # D_1(chi), for chi primitive, chi^2 != 1. # Defined for p >= 3. # A = <beta> (alpha is trivial) with ord(beta) = p-1. # All non-trivial characters are primitive. # Relevant character indices: [0, (1..(p-3)/2)]. for i in [1 .. (p-3)/2] do
rho := SL2IrrepD(p, 1, [0, i])[1];
name := Concatenation("D_1([0,", String(i), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; fi;
if p - 1 = degree then # N_1(chi), for chi primitive. # A = <zeta>, ord(zeta) = p+1. # Relevant character indices: [0, (1..(p-1)/2)]. for j in [1 .. (p-1)/2] do
rho := SL2IrrepN(p, 1, [0, j])[1];
name := Concatenation("N_1([0,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; fi;
if (p+1)/2 = degree then # R_1(1)+ and R_1(u)+, for u a non-residue. # SL2IrrepRUnary gives a list containing two reps, + and - .
rho := SL2IrrepRUnary(p, 1, 1)[1];
name := "R_1(1)+";
_SL2RecordIrrep(irrep_list, name, rho, l);
u := _SL2QuadNonRes(p);
rho := SL2IrrepRUnary(p, 1, u)[1];
name := Concatenation("R_1(", String(u), ")+");
_SL2RecordIrrep(irrep_list, name, rho, l); fi;
if (p-1)/2 = degree then # R_1(1)- and R_1(u)-, for u a non-residue. # SL2IrrepRUnary gives a list containing two reps, + and - . # For p=3, R_1(1)- = Xi_4 and R_1(2)- = Xi_8, the two linear reps. of level 3.
rho := SL2IrrepRUnary(p, 1, 1)[2];
name := "R_1(1)-";
_SL2RecordIrrep(irrep_list, name, rho, l);
u := _SL2QuadNonRes(p);
rho := SL2IrrepRUnary(p, 1, u)[2];
name := Concatenation("R_1(", String(u), ")-");
_SL2RecordIrrep(irrep_list, name, rho, l); fi;
if p = degree then # N_1(nu), the Steinberg representation.
rho := SL2IrrepN(p, 1, [0, 0])[1];
name := "N_1(nu)";
_SL2RecordIrrep(irrep_list, name, rho, l); fi;
od;
# p odd, ld >= 2.
pmax := Maximum([RootInt(2*degree+1, 2), 3]);
ldmax := Maximum([LogInt(degree, 3) + 2, 2]);
pset := Filtered([3..pmax], x -> IsPrime(x));
ldset := [2..ldmax]; for p in pset do for ld in ldset do
l := p^ld; if (p+1)*p^(ld-1) = degree then # D_ld(chi), for chi primitive, chi^2 != 1. # A = <alpha> x <beta>, with ord(alpha) = p^(ld-1) and ord(beta) = p-1. # chi is primitive iff it is injective on <alpha>. # Remember chi and bar(chi) give same rep; for j = 0 or (p-1)/2, # bar(chi(i,j)) = chi(-i, j). # We handle that case first: for i in PrimeResidues(p^(ld-1)) do if i > p^(ld-1) / 2then
break; fi;
rho := SL2IrrepD(p, ld, [i, 0])[1];
name := Concatenation("D_", String(ld), "([", String(i), ",0])");
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := SL2IrrepD(p, ld, [i, (p-1)/2])[1];
name := Concatenation("D_", String(ld), "([", String(i), ",", String((p-1)/2), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; # Then the rest: for i in PrimeResidues(p^(ld-1)) do for j in [1 .. (p-3)/2] do if Gcd(i, p) = 1then
rho := SL2IrrepD(p, ld, [i, j])[1];
name := Concatenation("D_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l); fi;
od;
od; fi;
if (p-1)*p^(ld-1) = degree then # N_ld(chi), for chi primitive. # A = <alpha> x <zeta>; ord(alpha) = p^(ld-1), ord(zeta) = p+1. # chi is primitive if injective on <alpha>. # Remember chi and bar(chi) give same rep; for j = 0 or (p+1)/2, # bar(chi(i,j)) = chi(-i, j). # We handle that case first: for i in PrimeResidues(p^(ld-1)) do if i > p^(ld-1) / 2then
break; fi;
rho := SL2IrrepN(p, ld, [i, 0])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",0])");
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := SL2IrrepN(p, ld, [i, (p+1)/2])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",", String((p+1)/2), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; # Then the rest: for i in PrimeResidues(p^(ld-1)) do for j in [1 .. (p-1)/2] do
rho := SL2IrrepN(p, ld, [i, j])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; fi;
if p^(ld-2)*(p^2-1)/2 = degree then # R_ld^si(r,t,chi), for 1 <= si <= ld-1, r,t either 1 or a non-residue, and chi primitive for si in [1 .. ld-1] do
u := _SL2QuadNonRes(p); for r in [1,u] do for t in [1,u] do if p = 3 and ld >= 3 and si = 1 and t = 1then # Special case, see RepR. # A = <alpha> x <zeta> where ord(alpha) = 3^(ld-2) and ord(zeta) = 6. # chi is primitive if injective on <alpha>. # Remember chi and bar(chi) give same rep; for j = 0 or 3, # bar(chi(i,j)) = chi(-i, j). for i in PrimeResidues(3^(ld-2)) do if i > 3^(ld-2) / 2then
break; fi; for j in [0,3] do
rho := SL2IrrepR(p, ld, si, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^", String(si), "(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for i in PrimeResidues(3^(ld-2)) do for j in [1,2] do
rho := SL2IrrepR(p, ld, si, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^", String(si), "(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; else # A is cyclic; A = <alpha> x <-1> where Ord(alpha) = p^(ld-si). # A character is primitive iff it is injective on <alpha>. for i in PrimeResidues(p^(ld-si)) do if i > p^(ld-si) / 2then
break; fi; for j in [0,1] do
rho := SL2IrrepR(p, ld, si, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^", String(si), "(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; fi;
od;
od;
od;
# (R_ld(1)+-)_1 and (R_ld(u)+-)_1, for u a non-residue
rho := SL2IrrepRUnary(p, ld, 1);
name := Concatenation("(R_", String(ld), "(1)+)_1");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("(R_", String(ld), "(1)-)_1");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
# p = 2, ld = 1.
l := 2; if1 = degree then # Xi_6 = C_2, with s = t = -1.
rho := [
[[-1]],
[[-1]]
];
name := "Xi_6";
_SL2RecordIrrep(irrep_list, name, rho, l); fi; if2 = degree then # N_1(nu), the Steinberg representation.
rho := SL2IrrepN(2, 1, [0, 0])[1];
name := "N_1(nu)";
_SL2RecordIrrep(irrep_list, name, rho, l); fi;
# p = 2, ld = 2.
l := 4; if3 = degree then # D_2(chi)+-, for chi != 1. Note that there is only one non-trivial character, # indexed by [1,0]. SL2IrrepD returns a list of the two subreps, + and - .
rho := SL2IrrepD(2, 2, [1, 0]);
name := "D_2([1,0])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "D_2([1,0])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
# R_2^0(1,3)_1 and C_2 tensor R_2^0(1,3)_1 (exceptional). # Matrix given in notes sec. 1.6.2.
w := Sqrt(2);
rho := [ 1/2 * [
[-1, 1, w],
[ 1, -1, w],
[ w, w, 0]
],
DiagonalMat([E(4), -E(4), 1])
];
name := "R_2^0(1,3)_1";
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := [-1*rho[1], -1*rho[2]];
name := "Xi_6 tensor R_2^0(1,3)_1";
_SL2RecordIrrep(irrep_list, name, rho, l); fi; if2 = degree then # N_2(chi), for chi primitive, chi^2 != 1. # A = <zeta> with ord(zeta) = 6. # For this specific case, chi is primitive iff chi(-1) = -1, which leaves only # a single relevant character, indexed by [0,1].
rho := SL2IrrepN(2, 2, [0, 1])[1];
name := "N_2([0,1])";
_SL2RecordIrrep(irrep_list, name, rho, l); fi; if1 = degree then # Xi_3 = C_4, with s = -1, t = i.
rho := [
[[-E(4)]],
[[E(4)]]
];
name := "Xi_3";
_SL2RecordIrrep(irrep_list, name, rho, l);
# Xi_9 = C_3, with s = i, t = -i.
rho := [
[[E(4)]],
[[-E(4)]]
];
name := "Xi_9";
_SL2RecordIrrep(irrep_list, name, rho, l); fi;
# p = 2, ld = 3.
l := 8; if6 = degree then # D_3(chi)+-, for chi primitive. # A = <5> x <-1>, and a character is primitive if injective on <5>. # This gives two characters, [1,0] and [1,1] (note that both square to 1). # SL2IrrepD returns a list of the two subreps, + and - .
rho := SL2IrrepD(2, 3, [1, 0]);
name := "D_3([1,0])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "D_3([1,0])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
rho := SL2IrrepD(2, 3, [1, 1]);
name := "D_3([1,1])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "D_3([1,1])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
rho := [E(4) * rho[1], -E(4) * rho[2]];
name := "Xi_9 tensor R_3^0(1,3,nu)_1";
_SL2RecordIrrep(irrep_list, name, rho, l); fi; if4 = degree then # N_3(chi), for chi primitive, chi^2 != 1. # A = <alpha> x <zeta> where ord(alpha) = 2 and ord(zeta) = 6. # chi is primitive if injective on <alpha>. # This gives two relevant characters, [1,1] and [1,2].
rho := SL2IrrepN(2, 3, [1, 1])[1];
name := "N_3([1,1])";
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := SL2IrrepN(2, 3, [1, 2])[1];
name := "N_3([1,2])";
_SL2RecordIrrep(irrep_list, name, rho, l); fi; if2 = degree then # N_3(chi)+-, for chi primitive, chi^2 = 1. # As above; relevant characters are those which square to 1: [1,0] and [1,3]. # SL2IrrepN returns a list of the two subreps, + and - .
rho := SL2IrrepN(2, 3, [1, 0]);
name := "N_3([1,0])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "N_3([1,0])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
rho := SL2IrrepN(2, 3, [1, 3]);
name := "N_3([1,3])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "N_3([1,3])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l); fi; if3 = degree then # R_3^0(r,t,chi), for chi(zeta) = i and r in {1,3} and t in {1,5}. # A = <zeta> with ord(zeta) = 4. # Specifically, zeta = (0,1) when t = 1, and zeta = (2,1) when t = 5. # The given character is indexed by [0,1]. for r in [1,3] do for t in [1,5] do
rho := SL2IrrepR(2, 3, 0, r, t, [0,1])[1];
name := Concatenation("R_3^0(", String(r), ",", String(t), ",[0,1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
# R_3^0(1, t, chi)_+-, for chi != 1 and t in {3,7}. # |A| = 2, so there's only one non-trivial chi, indexed by [0,1]. # chi squares to 1, giving two subreps. # SL2IrrepR returns a list with the two supreps, + and -. for t in [3,7] do
rho := SL2IrrepR(2,3,0,1,t,[0,1]);
name := Concatenation("R_3^0(1,", String(t), ",[0,1])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_3^0(1,", String(t), ",[0,1])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
od; fi;
# p = 2, ld = 4.
l := 16; if24 = degree then # D_4(chi), for chi primitive, chi^2 != 1. # A = <5> x <-1>, and a character is primitive if injective on <5>. # Note that 5 has order 4. # This gives two relevant characters, [1,0] and [1,1].
rho := SL2IrrepD(2, 4, [1, 0])[1];
name := "D_4([1,0])";
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := SL2IrrepD(2, 4, [1, 1])[1];
name := "D_4([1,1])";
_SL2RecordIrrep(irrep_list, name, rho, l); fi; if8 = degree then # N_4(chi), for chi primitive, chi^2 != 1. # A = <alpha> x <zeta> where ord(alpha) = 4 and ord(zeta) = 6. # chi is primitive if injective on <alpha>. # This gives six relevant characters: [1,0],[1,3],[1,1],[1,2],[3,1],[3,2]. for x in [[1,0],[1,3],[1,1],[1,2],[3,1],[3,2]] do
rho := SL2IrrepN(2, 4, x)[1];
name := Concatenation("N_4([", String(x[1]), ",", String(x[2]), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; fi; if6 = degree then # R_4^0(r,t,chi), for chi primitive with chi^2 != 1, r in {1,3}, t in {1,5}. # For t = 1, A = <alpha> x <zeta>, with ord(alpha) = 2 and ord(zeta) = 4, # and a character is primitive iff injective on alpha, so the only relevant # character is [1,1]. # For t = 5, A = <alpha> x <-1>, with ord(alpha) = 4; in this case a character # is primitive iff injective on <-alpha^2> (see NW p. 496), which means [1,0]. for r in [1,3] do
rho := SL2IrrepR(2, 4, 0, r, 1, [1, 1])[1];
name := Concatenation("R_4^0(", String(r), ",1,[1,1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; for r in [1,3] do
rho := SL2IrrepR(2, 4, 0, r, 5, [1, 0])[1];
name := Concatenation("R_4^0(", String(r), ",5,[1,0])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
# R_4^0(1,t,chi)+-, for chi primitive, t in {3,7}. # A = <alpha> x <-1>, with ord(alpha) = 2. There are therefore two primitive chars, # indexed by [1,0] and [1,1]. Both square to 1, giving two subreps. # SL2IrrepR returns a list of the two subreps, + and - . for t in [3,7] do
rho := SL2IrrepR(2, 4, 0, 1, t, [1,0]);
name := Concatenation("R_4^0(1,", String(t), ",[1,0])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_4^0(1,", String(t), ",[1,0])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
# R_4^2(r,t,chi) and C_2 tensor R_4^2(r,t,chi), for chi != 1, r,t in {1,3}. # A = {+-1}, so there is a single non-trivial character, indexed by [0,1]. # NOTE: C_2 tensor R_4^2(1,1,[0,1]) is iso to R_4^2(1,3,nu)_1, and # C_2 tensor R_4^2(3,1,[0,1]) is iso to R_4^2(3,3,nu)_1. # NW lists them under the latter name, so this is what we do here; see below. for r in [1,3] do
rho := SL2IrrepR(2,4,2,r,1,[0,1])[1];
name := Concatenation("R_4^2(", String(r), ",1,[0,1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
# R_4^2(r,3,nu)_1, for r in {1,3}. # Basis is given on NW p. 524. # See above. for r in [1,3] do
rho := SL2IrrepR(2,4,2,r,3,[0,0])[1];
name := Concatenation("R_4^2(", String(r), ",3,nu)_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; fi; if3 = degree then # R_4^0(r,t,chi)+-, for chi primitive, chi^2 = 1, r in {1,3}, t in {1,5}. # See previous. For t = 1, the relevant chars. are [1,0] and [1,2]. # For t = 5, the relevant chars. are [0,1] and [2,1]. # SL2IrrepR returns a list with the two supreps, + and -. for r in [1,3] do
rho := SL2IrrepR(2, 4, 0, r, 1, [1,0]);
name := Concatenation("R_4^0(", String(r), ",1,[1,0])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_4^0(", String(r), ",1,[1,0])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
rho := SL2IrrepR(2, 4, 0, r, 1, [1,2]);
name := Concatenation("R_4^0(", String(r), ",1,[1,2])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_4^0(", String(r), ",1,[1,2])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
od; for r in [1,3] do
rho := SL2IrrepR(2, 4, 0, r, 5, [0,1]);
name := Concatenation("R_4^0(", String(r), ",5,[0,1])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_4^0(", String(r), ",5,[0,1])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
rho := SL2IrrepR(2, 4, 0, r, 5, [2,1]);
name := Concatenation("R_4^0(", String(r), ",5,[2,1])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_4^0(", String(r), ",5,[2,1])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
od; fi; if12 = degree then # N_3(chi)+ tensor R_4^0(1,7,psi)+, for psi(alpha) = -1, psi(-1) = 1, # chi primitive and chi^2 = 1. # Calculated via Kronecker product. # For the N part, chars. are [1,0] and [1,3] (see the ld=3 section earlier). # # First construct R_4^0(1,7,psi)+. Char is [1,1]. SL2IrrepR gives a list with + and then -.
x := SL2IrrepR(2, 4, 0, 1, 7, [1, 1]); for j in [0,3] do # Construct N_3(chi)+. SL2IrrepR again returns a list.
rho := SL2IrrepN(2, 3, [1, j]);
rho := [
KroneckerProduct(rho[1][1], x[1][1]),
KroneckerProduct(rho[1][2], x[1][2]), 12
];
name := Concatenation("N_3([1,", String(j), "])+ tensor R_4^0(1,7,[1,1])+");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; fi;
# p = 2, ld = 5.
l := 32; if48 = degree then # D_5(chi), for chi primitive. # A = <5> x <-1>, and a character is primitive if injective on <5>. # Note that 5 has order 8. # This gives four relevant characters, [1,0], [1,1], [3,0], [3,1]. for i in [1,3] do for j in [0,1] do
rho := SL2IrrepD(2, 5, [i, j])[1];
name := Concatenation("D_5([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; fi; if16 = degree then # N_5(chi), for chi primitive. # A = <alpha> x <zeta> where ord(alpha) = 8 and ord(zeta) = 6. # chi is primitive if injective on <alpha>. for i in [1,3] do for j in [0,3] do
rho := SL2IrrepN(2, 5, [i,j])[1];
name := Concatenation("N_5([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for i in [1,3,5,7] do for j in [1,2] do
rho := SL2IrrepN(2, 5, [i,j])[1];
name := Concatenation("N_5([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; fi; if12 = degree then # R_5^0(r,t,chi), for chi primitive, r in {1,3}, t in {1,5}. # For t = 1, A = <alpha> x <zeta>, with ord(alpha) = 4 and ord(zeta) = 4. # For t = 5, A = <alpha> x <-1>, with ord(alpha) = 8. # In both cases, a character is primitive iff injective on alpha. for r in [1,3] do for j in [0,2] do
rho := SL2IrrepR(2, 5, 0, r, 1, [1,j])[1];
name := Concatenation("R_5^0(", String(r), ",1,[1,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; for i in [1,3] do
rho := SL2IrrepR(2, 5, 0, r, 1, [i,1])[1];
name := Concatenation("R_5^0(", String(r), ",1,[", String(i), ",1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for r in [1,3] do for i in [1,3] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 0, r, 5, [i,j])[1];
name := Concatenation("R_5^0(", String(r), ",5,[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
# R_5^1(r,t,chi), for chi primitive and: # r in {1,5}, t in {1,5}, or # r in {1,3}, t in {3,7}. # A = <alpha> x <-1>; ord(alpha) = 4. Prim. iff injective on alpha. # Chars are therefore [0,1] and [1,1]. for r in [1,5] do for t in [1,5] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 1, r, t, [1,j])[1];
name := Concatenation("R_5^1(", String(r), ",", String(t), ",[1,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od; for r in [1,3] do for t in [3,7] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 1, r, t, [1,j])[1];
name := Concatenation("R_5^1(", String(r), ",", String(t), ",[1,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
# R_5^2(r,1,chi)_1 and C_3 tensor R_5^2(r,1,chi)_1, # for chi NOT primitive and r in {1,3}. # A = <alpha> x <-1>, so non-primitive characters are indexed by [0,0] and [0,1]. for r in [1,3] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 2, r, 1, [0,j])[1];
name := Concatenation("R_5^2(", String(r), ",1,[0,", String(j), "])_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := [E(4) * rho[1], -E(4) * rho[2], 12];
name := Concatenation("Xi_9 tensor R_5^2(", String(r), ",1,[0,", String(j), "])_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; fi; if24 = degree then # R_5^0(1,t,chi), for chi primitive and t in {3,7}. # A = <alpha> x <-1> with ord(alpha) = 4. # A character is primitive iff injective on <alpha>. # Relevant chars. are therefore [1,0] and [1,1]. for t in [3,7] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 0, 1, t, [1,j])[1];
name := Concatenation("R_5^0(1,", String(t), ",[1,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; fi; if6 = degree then # R_5^2(r,t,chi)+-, for chi primitive, r in {1,3}, and t in {1,3,5,7}. # A = <alpha> x <-1>, with ord(alpha) = 2. # Relevant chars are therefore [1,0] and [1,1], both of which square to 1. # SL2IrrepR returns a list of the two resulting subreps, + and - . for r in [1,3] do for t in [1,3,5,7] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 2, r, t, [1,j]);
name := Concatenation("R_5^2(", String(r), ",", String(t), ",[1,", String(j), "])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_5^2(", String(r), ",", String(t), ",[1,", String(j), "])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
od;
od;
od; fi;
# p = 2, ld > 5.
ldmax := Maximum([LogInt(degree, 2) + 4, 6]);
ldset := [6..ldmax]; for ld in ldset do
l := 2^ld; if3*2^(ld-1) = degree then # D_ld(chi), for chi primitive. # A = <5> x <-1>, and a character is primitive if injective on <5>. # Note that 5 has order 2^(ld-2). for i in PrimeResidues(2^(ld-2) / 2) do for j in [0,1] do
rho := SL2IrrepD(2, ld, [i, j])[1];
name := Concatenation("D_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; fi; if2^(ld-1) = degree then # N_ld(chi), for chi primitive. # A = <alpha> x <zeta> where ord(alpha) = 2^(ld-2) and ord(zeta) = 6. # chi is primitive if injective on <alpha>. for i in PrimeResidues(2^(ld-2) / 2) do for j in [0,3] do
rho := SL2IrrepN(2, ld, [i,j])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for i in PrimeResidues(2^(ld-2)) do for j in [1,2] do
rho := SL2IrrepN(2, ld, [i,j])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; fi; if3*2^(ld-2) = degree then # R_ld^0(1,t,chi), for chi primitive and t in {3,7}. # A = <alpha> x <-1> with ord(alpha) = 2^(ld-3). # A character is primitive iff injective on <alpha>. for t in [3,7] do for i in PrimeResidues(2^(ld-3) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 0, 1, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^0(1,", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od; fi; if3*2^(ld-3) = degree then # R_ld^si(r,t,chi), for chi primitive and: # si = 0, r in {1,3}, t = 1 (ord(alpha) = 2^(ld-3), ord(zeta) = 4), # si = 0, r in {1,3}, t = 5 (ord(alpha) = 2^(ld-2), ord(zeta) = 2), # si = 1, r in {1,5}, t in {1,5} or r in {1,3}, t in {3,7} (ord(alpha) = 2^(ld-3), ord(zeta) = 2), # si = 2, r in {1,3}, t in {1,3,5,7} (ord(alpha) = 2^(ld-4), ord(zeta) = 2). # In all cases, a character is primitive iff injective on <alpha>. for r in [1,3] do for i in PrimeResidues(2^(ld-3) / 2) do for j in [0,2] do
rho := SL2IrrepR(2, ld, 0, r, 1, [i,j])[1];
name := Concatenation("R_", String(ld), "^0(", String(r), ",1,[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for i in PrimeResidues(2^(ld-3)) do
rho := SL2IrrepR(2, ld, 0, r, 1, [i,1])[1];
name := Concatenation("R_", String(ld), "^0(", String(r), ",1,[", String(i), ",1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for r in [1,3] do for i in PrimeResidues(2^(ld-2) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 0, r, 5, [i,j])[1];
name := Concatenation("R_", String(ld), "^0(", String(r), ",5,[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od; for r in [1,5] do for t in [1,5] do for i in PrimeResidues(2^(ld-3) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 1, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^1(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
od; for r in [1,3] do for t in [3,7] do for i in PrimeResidues(2^(ld-3) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 1, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^1(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
od; for r in [1,3] do for t in [1,3,5,7] do for i in PrimeResidues(2^(ld-4) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 2, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^2(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
od; fi; if3*2^(ld-4) = degree then # R_ld^si(r,t,chi), for chi primitive, si in {3, ..., ld-3}, and r,t in {1,3,5,7}. # ord(alpha) = 2^(ld-si-1), ord(zeta) = 2. # A character is primitive iff injective on <alpha>. for si in [3 .. (ld-3)] do for r in [1,3,5,7] do for t in [1,3,5,7] do for i in PrimeResidues(2^(ld-si-1) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, si, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^", String(si), "(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
od;
od;
# R_ld^(ld-2)(r,t,chi), for chi primitive, r in {1,3,5,7}, t in {1,3}. # A = <alpha> x <-1>, with ord(alpha) = 2. # A character is primitive iff injective on alpha (but note alternate def. # of primitive; see NW p. 496). Two such exist, indexed by [1,0] and [1,1]. for r in [1,3,5,7] do for t in [1,3] do
rho := SL2IrrepR(2, ld, ld-2, r, t, [1,0])[1];
name := Concatenation("R_", String(ld), "^", String(ld-2), "(", String(r), ",", String(t), ",[1,0])");
_SL2RecordIrrep(irrep_list, name, rho, l);
if ld = 6then # R_6^4(r,t,nu)_1 and C_2 tensor R_6^4(r,t,nu)_1, for r in {1,3,5,7}, t in {1,3}. for r in [1,3,5,7] do for t in [1,3] do
rho := SL2IrrepR(2,6,4,r,t,[0,0])[1];
name := Concatenation("R_6^4(", String(r), ",", String(t), ",nu)_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := [-1 * rho[1], -1 * rho[2]];
name := Concatenation("Xi_6 tensor R_6^4(", String(r), ",", String(t), ",nu)_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; else # R_ld^(ld-3)(r,t,chi)_1, for chi in {[0,0],[2,0]} (which both square to 1), # r in {1,3,5,7}, t in {1,3}. for r in [1,3,5,7] do for t in [1,3] do
rho := SL2IrrepR(2,ld,ld-3,r,t,[0,0])[1];
name := Concatenation("R_", String(ld), "^", String(ld-3), "(", String(r), ",", String(t), ",nu)_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
Info(InfoSL2Reps, 1, "SL2Reps : ", Length(irrep_list), " irreps of degree ", degree, " found.");
return irrep_list; end );
InstallGlobalFunction( _SL2IrrepsPPLOfMaxDegree, function(max_degree)
local output, degree, count;
if not max_degree in PositiveIntegers then
Error("max_degree must be a positive integer."); fi;
output := [];
count := 0;
for degree in [1..max_degree] do
Info(InfoSL2Reps, 1, "SL2Reps : Degree ", degree, ":");
output[degree] := _SL2IrrepsPPLOfDegree(degree);
count := count + Length(output[degree]);
od;
Info(InfoSL2Reps, 1, "SL2Reps : ", count, " total irreps found.");
return output; end );
InstallGlobalFunction( _SL2IrrepsPPLOfLevel, function(p, ld)
local irrep_list, l, si, r, t, u, w, x, i, j, name, rho;
if not IsPrime(p) then
Error("p must be a prime."); elif (not ld in Integers) or (ld < 0) then
Error("ld must be a non-negative integer."); fi;
l := p^ld;
irrep_list := [];
if ld = 0then # The trivial irrep Xi_0 = C_1.
_SL2RecordIrrep(irrep_list, "Xi_0", [[[1]], [[1]]], 1); elif p > 2then if ld = 1then # D_1(chi), for chi primitive, chi^2 != 1. # Defined for p >= 3. # A = <beta> (alpha is trivial) with ord(beta) = p-1. # All non-trivial characters are primitive. # Relevant character indices: [0, (1..(p-3)/2)]. for i in [1 .. (p-3)/2] do
rho := SL2IrrepD(p, 1, [0, i])[1];
name := Concatenation("D_1([0,", String(i), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
# N_1(chi), for chi primitive. # A = <zeta>, ord(zeta) = p+1. # Relevant character indices: [0, (1..(p-1)/2)]. for j in [1 .. (p-1)/2] do
rho := SL2IrrepN(p, 1, [0, j])[1];
name := Concatenation("N_1([0,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
# R_1(1)+- and R_1(u)+-, for u a non-residue. # SL2IrrepRUnary gives a list containing two reps, + and - . # For p=3, R_1(1)- = Xi_4 and R_1(2)- = Xi_8, the two linear reps. of level 3.
rho := SL2IrrepRUnary(p, 1, 1);
name := "R_1(1)+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "R_1(1)-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
u := _SL2QuadNonRes(p);
rho := SL2IrrepRUnary(p, 1, u);
name := Concatenation("R_1(", String(u), ")+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_1(", String(u), ")-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
# D_ld(chi), for chi primitive, chi^2 != 1. # A = <alpha> x <beta>, with ord(alpha) = p^(ld-1) and ord(beta) = p-1. # chi is primitive iff it is injective on <alpha>. # Remember chi and bar(chi) give same rep; for j = 0 or (p-1)/2, # bar(chi(i,j)) = chi(-i, j). # We handle that case first: for i in PrimeResidues(p^(ld-1)) do if i > p^(ld-1) / 2then
break; fi;
rho := SL2IrrepD(p, ld, [i, 0])[1];
name := Concatenation("D_", String(ld), "([", String(i), ",0])");
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := SL2IrrepD(p, ld, [i, (p-1)/2])[1];
name := Concatenation("D_", String(ld), "([", String(i), ",", String((p-1)/2), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; # Then the rest: for i in PrimeResidues(p^(ld-1)) do for j in [1 .. (p-3)/2] do if Gcd(i, p) = 1then
rho := SL2IrrepD(p, ld, [i, j])[1];
name := Concatenation("D_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l); fi;
od;
od;
# N_ld(chi), for chi primitive. # A = <alpha> x <zeta>; ord(alpha) = p^(ld-1), ord(zeta) = p+1. # chi is primitive if injective on <alpha>. # Remember chi and bar(chi) give same rep; for j = 0 or (p+1)/2, # bar(chi(i,j)) = chi(-i, j). for i in PrimeResidues(p^(ld-1)) do if i > p^(ld-1) / 2then
break; fi;
rho := SL2IrrepN(p, ld, [i, 0])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",0])");
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := SL2IrrepN(p, ld, [i, (p+1)/2])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",", String((p+1)/2), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; for i in PrimeResidues(p^(ld-1)) do for j in [1 .. (p-1)/2] do
rho := SL2IrrepN(p, ld, [i, j])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
# R_ld^si(r,t,chi), for 1 <= si <= ld-1, r,t either 1 or a non-residue, and chi primitive for si in [1 .. ld-1] do
u := _SL2QuadNonRes(p); for r in [1,u] do for t in [1,u] do if p = 3 and ld >= 3 and si = 1 and t = 1then # Special case, see RepRs. # A = <alpha> x <zeta> where ord(alpha) = 3^(ld-2) and ord(zeta) = 6. # chi is primitive if injective on <alpha>. # Remember chi and bar(chi) give same rep; for j = 0 or 3, # bar(chi(i,j)) = chi(-i, j). for i in PrimeResidues(3^(ld-2)) do if i > 3^(ld-2) / 2then
break; fi; for j in [0,3] do
rho := SL2IrrepR(p, ld, si, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^", String(si), "(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for i in PrimeResidues(3^(ld-2)) do for j in [1,2] do
rho := SL2IrrepR(p, ld, si, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^", String(si), "(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; else # A is cyclic; A = <alpha> x <-1> where Ord(alpha) = p^(ld-si). # A character is primitive iff it is injective on <alpha>. for i in PrimeResidues(p^(ld-si)) do if i > p^(ld-si) / 2then
break; fi; for j in [0,1] do
rho := SL2IrrepR(p, ld, si, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^", String(si), "(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; fi;
od;
od;
od;
# (R_ld(1)+-)_1 and (R_ld(u)+-)_1, for u a non-residue
rho := SL2IrrepRUnary(p, ld, 1);
name := Concatenation("(R_", String(ld), "(1)+)_1");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("(R_", String(ld), "(1)-)_1");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
u := _SL2QuadNonRes(p);
rho := SL2IrrepRUnary(p, ld, u);
name := Concatenation("(R_", String(ld), "(", String(u), ")+)_1");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("(R_", String(ld), "(", String(u), ")-)_1");
_SL2RecordIrrep(irrep_list, name, rho[2], l); fi; else # p = 2 if ld = 1then # Xi_6 = C_2, with s = t = -1.
rho := [
[[-1]],
[[-1]]
];
name := "Xi_6";
_SL2RecordIrrep(irrep_list, name, rho, l);
# N_1(nu), the Steinberg representation.
rho := SL2IrrepN(2, 1, [0, 0])[1];
name := "N_1(nu)";
_SL2RecordIrrep(irrep_list, name, rho, l); elif ld = 2then # D_2(chi)+-, for chi != 1. Note that there is only one non-trivial character, # indexed by [1,0]. SL2IrrepD returns a list of the two subreps, + and - .
rho := SL2IrrepD(2, 2, [1, 0]);
name := "D_2([1,0])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "D_2([1,0])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
# R_2^0(1,3)_1 and C_2 tensor R_2^0(1,3)_1 (exceptional). # Matrix given in notes sec. 1.6.2.
w := Sqrt(2);
rho := [ 1/2 * [
[-1, 1, w],
[ 1, -1, w],
[ w, w, 0]
],
DiagonalMat([E(4), -E(4), 1])
];
name := "R_2^0(1,3)_1";
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := [-1*rho[1], -1*rho[2]];
name := "Xi_6 tensor R_2^0(1,3)_1";
_SL2RecordIrrep(irrep_list, name, rho, l);
# N_2(chi), for chi primitive, chi^2 != 1. # A = <zeta> with ord(zeta) = 6. # For this specific case, chi is primitive iff chi(-1) = -1, which leaves only # a single relevant character, indexed by [0,1].
rho := SL2IrrepN(2, 2, [0, 1])[1];
name := "N_2([0,1])";
_SL2RecordIrrep(irrep_list, name, rho, l);
# Xi_3 = C_4, with s = -1, t = i.
rho := [
[[-E(4)]],
[[E(4)]]
];
name := "Xi_3";
_SL2RecordIrrep(irrep_list, name, rho, l);
# Xi_9 = C_3, with s = i, t = -i.
rho := [
[[E(4)]],
[[-E(4)]]
];
name := "Xi_9";
_SL2RecordIrrep(irrep_list, name, rho, l); elif ld = 3then # D_3(chi)+-, for chi primitive. # A = <5> x <-1>, and a character is primitive if injective on <5>. # This gives two characters, [1,0] and [1,1] (note that both square to 1). # SL2IrrepD returns a list of the two subreps, + and - .
rho := SL2IrrepD(2, 3, [1, 0]);
name := "D_3([1,0])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "D_3([1,0])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
rho := SL2IrrepD(2, 3, [1, 1]);
name := "D_3([1,1])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "D_3([1,1])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
# N_3(chi), for chi primitive, chi^2 != 1. # A = <alpha> x <zeta> where ord(alpha) = 2 and ord(zeta) = 6. # chi is primitive if injective on <alpha>. # This gives two relevant characters, [1,1] and [1,2].
rho := SL2IrrepN(2, 3, [1, 1])[1];
name := "N_3([1,1])";
_SL2RecordIrrep(irrep_list, name, rho, l);
# N_3(chi)+-, for chi primitive, chi^2 = 1. # As above; relevant characters are those which square to 1: [1,0] and [1,3]. # SL2IrrepN returns a list of the two subreps, + and - .
rho := SL2IrrepN(2, 3, [1, 0]);
name := "N_3([1,0])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "N_3([1,0])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
rho := SL2IrrepN(2, 3, [1, 3]);
name := "N_3([1,3])+";
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := "N_3([1,3])-";
_SL2RecordIrrep(irrep_list, name, rho[2], l);
# R_3^0(r,t,chi), for chi(zeta) = i and r in {1,3} and t in {1,5}. # A = <zeta> with ord(zeta) = 4. # Specifically, zeta = (0,1) when t = 1, and zeta = (2,1) when t = 5. # The given character is indexed by [0,1]. for r in [1,3] do for t in [1,5] do
rho := SL2IrrepR(2, 3, 0, r, t, [0,1])[1];
name := Concatenation("R_3^0(", String(r), ",", String(t), ",[0,1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
# R_3^0(1, t, chi)_+-, for chi != 1 and t in {3,7}. # |A| = 2, so there's only one non-trivial chi, indexed by [0,1]. # chi squares to 1, giving two subreps. # SL2IrrepR returns a list with the two subreps, + and -. for t in [3,7] do
rho := SL2IrrepR(2,3,0,1,t,[0,1]);
name := Concatenation("R_3^0(1,", String(t), ",[0,1])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_3^0(1,", String(t), ",[0,1])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
od; elif ld = 4then # D_4(chi), for chi primitive, chi^2 != 1. # A = <5> x <-1>, and a character is primitive if injective on <5>. # Note that 5 has order 4. # This gives two relevant characters, [1,0] and [1,1].
rho := SL2IrrepD(2, 4, [1, 0])[1];
name := "D_4([1,0])";
_SL2RecordIrrep(irrep_list, name, rho, l);
# N_4(chi), for chi primitive, chi^2 != 1. # A = <alpha> x <zeta> where ord(alpha) = 4 and ord(zeta) = 6. # chi is primitive if injective on <alpha>. # This gives six relevant characters: [1,0],[1,3],[1,1],[1,2],[3,1],[3,2]. for x in [[1,0],[1,3],[1,1],[1,2],[3,1],[3,2]] do
rho := SL2IrrepN(2, 4, x)[1];
name := Concatenation("N_4([", String(x[1]), ",", String(x[2]), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
# R_4^0(r,t,chi), for chi primitive with chi^2 != 1, r in {1,3}, t in {1,5}. # For t = 1, A = <alpha> x <zeta>, with ord(alpha) = 2 and ord(zeta) = 4, # and a character is primitive iff injective on alpha, so the only relevant # character is [1,1]. # For t = 5, A = <alpha> x <-1>, with ord(alpha) = 4; in this case a character # is primitive iff injective on <-alpha^2> (see NW p. 496), which means [1,0]. for r in [1,3] do
rho := SL2IrrepR(2, 4, 0, r, 1, [1, 1])[1];
name := Concatenation("R_4^0(", String(r), ",1,[1,1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; for r in [1,3] do
rho := SL2IrrepR(2, 4, 0, r, 5, [1, 0])[1];
name := Concatenation("R_4^0(", String(r), ",5,[1,0])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
# R_4^0(1,t,chi)+-, for chi primitive, t in {3,7}. # A = <alpha> x <-1>, with ord(alpha) = 2. There are therefore two primitive chars, # indexed by [1,0] and [1,1]. Both square to 1, giving two subreps. # SL2IrrepR returns a list of the two subreps, + and - . for t in [3,7] do
rho := SL2IrrepR(2, 4, 0, 1, t, [1,0]);
name := Concatenation("R_4^0(1,", String(t), ",[1,0])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_4^0(1,", String(t), ",[1,0])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
# R_4^2(r,t,chi) and C_2 tensor R_4^2(r,t,chi), for chi != 1, r,t in {1,3}. # A = {+-1}, so there is a single non-trivial character, indexed by [0,1]. # NOTE: C_2 tensor R_4^2(1,1,[0,1]) is iso to R_4^2(1,3,nu)_1, and # C_2 tensor R_4^2(3,1,[0,1]) is iso to R_4^2(3,3,nu)_1. # NW lists them under the latter name, so this is what we do here; see below. for r in [1,3] do
rho := SL2IrrepR(2,4,2,r,1,[0,1])[1];
name := Concatenation("R_4^2(", String(r), ",1,[0,1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
# R_4^2(r,3,nu)_1, for r in {1,3}. # Basis is given on NW p. 524. # See above. for r in [1,3] do
rho := SL2IrrepR(2,4,2,r,3,[0,0])[1];
name := Concatenation("R_4^2(", String(r), ",3,nu)_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
# R_4^0(r,t,chi)+-, for chi primitive, chi^2 = 1, r in {1,3}, t in {1,5}. # See previous. For t = 1, the relevant chars. are [1,0] and [1,2]. # For t = 5, the relevant chars. are [0,1] and [2,1]. # SL2IrrepR returns a list with the two supreps, + and -. for r in [1,3] do
rho := SL2IrrepR(2, 4, 0, r, 1, [1,0]);
name := Concatenation("R_4^0(", String(r), ",1,[1,0])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_4^0(", String(r), ",1,[1,0])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
rho := SL2IrrepR(2, 4, 0, r, 1, [1,2]);
name := Concatenation("R_4^0(", String(r), ",1,[1,2])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_4^0(", String(r), ",1,[1,2])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
od; for r in [1,3] do
rho := SL2IrrepR(2, 4, 0, r, 5, [0,1]);
name := Concatenation("R_4^0(", String(r), ",5,[0,1])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_4^0(", String(r), ",5,[0,1])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
# N_3(chi)+ tensor R_4^0(1,7,psi)+, for psi(alpha) = -1, psi(-1) = 1, # chi primitive and chi^2 = 1. # Calculated via Kronecker product. # For the N part, chars. are [1,0] and [1,3] (see the ld=3 section earlier). # # First construct R_4^0(1,7,psi)+. Char is [1,1]. SL2IrrepR gives a list with + and then -.
x := SL2IrrepR(2, 4, 0, 1, 7, [1, 1]); for j in [0,3] do # Construct N_3(chi)+. SL2IrrepR again returns a list.
rho := SL2IrrepN(2, 3, [1, j]);
rho := [
KroneckerProduct(rho[1][1], x[1][1]),
KroneckerProduct(rho[1][2], x[1][2]), 12
];
name := Concatenation("N_3([1,", String(j), "])+ tensor R_4^0(1,7,[1,1])+");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; elif ld = 5then # D_5(chi), for chi primitive. # A = <5> x <-1>, and a character is primitive if injective on <5>. # Note that 5 has order 8. # This gives four relevant characters, [1,0], [1,1], [3,0], [3,1]. for i in [1,3] do for j in [0,1] do
rho := SL2IrrepD(2, 5, [i, j])[1];
name := Concatenation("D_5([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
# N_5(chi), for chi primitive. # A = <alpha> x <zeta> where ord(alpha) = 8 and ord(zeta) = 6. # chi is primitive if injective on <alpha>. for i in [1,3] do for j in [0,3] do
rho := SL2IrrepN(2, 5, [i,j])[1];
name := Concatenation("N_5([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for i in [1,3,5,7] do for j in [1,2] do
rho := SL2IrrepN(2, 5, [i,j])[1];
name := Concatenation("N_5([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
# R_5^0(r,t,chi), for chi primitive, r in {1,3}, t in {1,5}. # For t = 1, A = <alpha> x <zeta>, with ord(alpha) = 4 and ord(zeta) = 4. # For t = 5, A = <alpha> x <-1>, with ord(alpha) = 8. # In both cases, a character is primitive iff injective on alpha. for r in [1,3] do for j in [0,2] do
rho := SL2IrrepR(2, 5, 0, r, 1, [1,j])[1];
name := Concatenation("R_5^0(", String(r), ",1,[1,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od; for i in [1,3] do
rho := SL2IrrepR(2, 5, 0, r, 1, [i,1])[1];
name := Concatenation("R_5^0(", String(r), ",1,[", String(i), ",1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for r in [1,3] do for i in [1,3] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 0, r, 5, [i,j])[1];
name := Concatenation("R_5^0(", String(r), ",5,[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
# R_5^1(r,t,chi), for chi primitive and: # r in {1,5}, t in {1,5}, or # r in {1,3}, t in {3,7}. # A = <alpha> x <-1>; ord(alpha) = 4. Prim. iff injective on alpha. # Chars are therefore [0,1] and [1,1]. for r in [1,5] do for t in [1,5] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 1, r, t, [1,j])[1];
name := Concatenation("R_5^1(", String(r), ",", String(t), ",[1,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od; for r in [1,3] do for t in [3,7] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 1, r, t, [1,j])[1];
name := Concatenation("R_5^1(", String(r), ",", String(t), ",[1,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
# R_5^2(r,1,chi)_1 and C_3 tensor R_5^2(r,1,chi)_1, # for chi NOT primitive and r in {1,3}. # A = <alpha> x <-1>, so non-primitive characters are indexed by [0,0] and [0,1]. for r in [1,3] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 2, r, 1, [0,j])[1];
name := Concatenation("R_5^2(", String(r), ",1,[0,", String(j), "])_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
# R_5^0(1,t,chi), for chi primitive and t in {3,7}. # A = <alpha> x <-1> with ord(alpha) = 4. # A character is primitive iff injective on <alpha>. # Relevant chars. are therefore [1,0] and [1,1]. for t in [3,7] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 0, 1, t, [1,j])[1];
name := Concatenation("R_5^0(1,", String(t), ",[1,", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
# R_5^2(r,t,chi)+-, for chi primitive, r in {1,3}, and t in {1,3,5,7}. # A = <alpha> x <-1>, with ord(alpha) = 2. # Relevant chars are therefore [1,0] and [1,1], both of which square to 1. # SL2IrrepR returns a list of the two resulting subreps, + and - . for r in [1,3] do for t in [1,3,5,7] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 2, r, t, [1,j]);
name := Concatenation("R_5^2(", String(r), ",", String(t), ",[1,", String(j), "])+");
_SL2RecordIrrep(irrep_list, name, rho[1], l);
name := Concatenation("R_5^2(", String(r), ",", String(t), ",[1,", String(j), "])-");
_SL2RecordIrrep(irrep_list, name, rho[2], l);
od;
od;
od; else # ld > 5
# D_ld(chi), for chi primitive. # A = <5> x <-1>, and a character is primitive if injective on <5>. # Note that 5 has order 2^(ld-2). for i in PrimeResidues(2^(ld-2) / 2) do for j in [0,1] do
rho := SL2IrrepD(2, ld, [i, j])[1];
name := Concatenation("D_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
# N_ld(chi), for chi primitive. # A = <alpha> x <zeta> where ord(alpha) = 2^(ld-2) and ord(zeta) = 6. # chi is primitive if injective on <alpha>. for i in PrimeResidues(2^(ld-2) / 2) do for j in [0,3] do
rho := SL2IrrepN(2, ld, [i,j])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for i in PrimeResidues(2^(ld-2)) do for j in [1,2] do
rho := SL2IrrepN(2, ld, [i,j])[1];
name := Concatenation("N_", String(ld), "([", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
# R_ld^0(1,t,chi), for chi primitive and t in {3,7}. # A = <alpha> x <-1> with ord(alpha) = 2^(ld-3). # A character is primitive iff injective on <alpha>. for t in [3,7] do for i in PrimeResidues(2^(ld-3) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 0, 1, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^0(1,", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
# R_ld^si(r,t,chi), for chi primitive and: # si = 0, r in {1,3}, t = 1 (ord(alpha) = 2^(ld-3), ord(zeta) = 4), # si = 0, r in {1,3}, t = 5 (ord(alpha) = 2^(ld-2), ord(zeta) = 2), # si = 1, r in {1,5}, t in {1,5} or r in {1,3}, t in {3,7} (ord(alpha) = 2^(ld-3), ord(zeta) = 2), # si = 2, r in {1,3}, t in {1,3,5,7} (ord(alpha) = 2^(ld-4), ord(zeta) = 2). # In all cases, a character is primitive iff injective on <alpha>. for r in [1,3] do for i in PrimeResidues(2^(ld-3) / 2) do for j in [0,2] do
rho := SL2IrrepR(2, ld, 0, r, 1, [i,j])[1];
name := Concatenation("R_", String(ld), "^0(", String(r), ",1,[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for i in PrimeResidues(2^(ld-3)) do
rho := SL2IrrepR(2, ld, 0, r, 1, [i,1])[1];
name := Concatenation("R_", String(ld), "^0(", String(r), ",1,[", String(i), ",1])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; for r in [1,3] do for i in PrimeResidues(2^(ld-2) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 0, r, 5, [i,j])[1];
name := Concatenation("R_", String(ld), "^0(", String(r), ",5,[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od; for r in [1,5] do for t in [1,5] do for i in PrimeResidues(2^(ld-3) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 1, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^1(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
od; for r in [1,3] do for t in [3,7] do for i in PrimeResidues(2^(ld-3) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 1, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^1(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
od; for r in [1,3] do for t in [1,3,5,7] do for i in PrimeResidues(2^(ld-4) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, 2, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^2(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
od;
# R_ld^si(r,t,chi), for chi primitive, si in {3, ..., ld-3}, and r,t in {1,3,5,7}. # ord(alpha) = 2^(ld-si-1), ord(zeta) = 2. # A character is primitive iff injective on <alpha>. for si in [3 .. (ld-3)] do for r in [1,3,5,7] do for t in [1,3,5,7] do for i in PrimeResidues(2^(ld-si-1) / 2) do for j in [0,1] do
rho := SL2IrrepR(2, ld, si, r, t, [i,j])[1];
name := Concatenation("R_", String(ld), "^", String(si), "(", String(r), ",", String(t), ",[", String(i), ",", String(j), "])");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od;
od;
od;
od;
# R_ld^(ld-2)(r,t,chi), for chi primitive, r in {1,3,5,7}, t in {1,3}. # A = <alpha> x <-1>, with ord(alpha) = 2. # A character is primitive iff injective on alpha (but note alternate def. # of primitive; see NW p. 496). Two such exist, indexed by [1,0] and [1,1]. for r in [1,3,5,7] do for t in [1,3] do
rho := SL2IrrepR(2, ld, ld-2, r, t, [1,0])[1];
name := Concatenation("R_", String(ld), "^", String(ld-2), "(", String(r), ",", String(t), ",[1,0])");
_SL2RecordIrrep(irrep_list, name, rho, l);
if ld = 6then # R_6^4(r,t,nu)_1 and C_2 tensor R_6^4(r,t,nu)_1, for r in {1,3,5,7}, t in {1,3}. for r in [1,3,5,7] do for t in [1,3] do
rho := SL2IrrepR(2,6,4,r,t,[0,0])[1];
name := Concatenation("R_6^4(", String(r), ",", String(t), ",nu)_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
rho := [-1 * rho[1], -1 * rho[2]];
name := Concatenation("Xi_6 tensor R_6^4(", String(r), ",", String(t), ",nu)_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
od;
od; else # R_ld^(ld-3)(r,t,chi)_1, for chi in {[0,0],[2,0]} (which both square to 1), # r in {1,3,5,7}, t in {1,3}. for r in [1,3,5,7] do for t in [1,3] do
rho := SL2IrrepR(2,ld,ld-3,r,t,[0,0])[1];
name := Concatenation("R_", String(ld), "^", String(ld-3), "(", String(r), ",", String(t), ",nu)_1");
_SL2RecordIrrep(irrep_list, name, rho, l);
Info(InfoSL2Reps, 1, "SL2Reps : ", Length(irrep_list), " irreps of level ", l, " found.");
return irrep_list; end );
#
InstallGlobalFunction( SL2IrrepsOfDegree, function(degree)
local linears, prime_power_reps, i, ConstructIrreps, triv, factorizations, output, f;
if not degree in PositiveIntegers then
Error("degree must be a positive integer."); fi;
prime_power_reps := [];
# collect prime-power-level irreps of degree dividing the given degree
Info(InfoSL2Reps, 1, "SL2Reps : Constructing irreps of prime-power level.");
Info(InfoSL2Reps, 1, "SL2Reps : Degree 1:");
# The linear reps are denoted Xi_n, n in Z/12Z, with T = [zeta_12^n] and S = [i^n]. # We handle these separately just for brevity in the output.
prime_power_reps[1] := [];
Info(InfoSL2Reps, 1, "SL2Reps : 12 irreps of degree 1 found.");
for i in DivisorsInt(degree) do if i = 1then
continue; fi;
Info(InfoSL2Reps, 1, "SL2Reps : Degree ", i, ":");
prime_power_reps[i] := _SL2IrrepsPPLOfDegree(i);
od;
ConstructIrreps := function(rho, factors, start)
local i, eta, output, name, new_start;
if Length(factors) = 0then # Nothing left to do.
return [rho]; fi;
output := [];
# Tensor the given rep., rho, with all coprime irreps of degree (factors[1]) # and recurse. for i in [start .. Length(prime_power_reps[factors[1]])] do
eta := prime_power_reps[factors[1]][i];
if Gcd(eta.level, rho.level) = 1then if Length(factors) > 1 and factors[2] = factors[1] then # Next degree is same as current. Need to keep irreps in order.
new_start := i+1; else
new_start := 1; fi;
output := Concatenation(
output,
ConstructIrreps(
rec(
S := KroneckerProduct(rho.S, eta.S),
T := KroneckerProduct(rho.T, eta.T),
degree := rho.degree * eta.degree,
level := rho.level * eta.level, # level is Lcm(level(rho), level(eta)), but they're coprime.
name := _SL2ConcatNames(rho.name, eta.name)
),
factors{[2..Length(factors)]},
new_start
)
); fi;
od;
return output; end;
triv := rec(
S := [[1]],
T := [[1]],
degree := 1,
level := 1,
name := "Xi_0"
);
factorizations := _SL2Factorizations(degree);
output := []; for f in factorizations do
Append(output, ConstructIrreps(triv, f, 1));
od;
# Sort by level.
SortBy(output, x -> x.level);
for i in [1 .. Length(output)] do
Info(InfoSL2Reps, 1, "SL2Reps : ", i, ": ( ", output[i].name, " ) [d: ", output[i].degree, ", l: ", output[i].level ,"]");
od;
Info(InfoSL2Reps, 1, "SL2Reps : Total count: ", Length(output));
return output; end );
InstallGlobalFunction( SL2IrrepsOfMaxDegree, function(max_degree)
local output, degree;
if not max_degree in PositiveIntegers then
Error("max_degree must be a positive integer."); fi;
output := [];
for degree in [1..max_degree] do
Info(InfoSL2Reps, 1, "SL2Reps : Degree ", degree, ":");
Append(output, SL2IrrepsOfDegree(degree));
od;
Info(InfoSL2Reps, 1, "SL2Reps : ", Length(output), " total irreps found.");
return output; end );
InstallGlobalFunction( SL2IrrepsOfLevel, function(l)
local output, pp_lists, factors, i, rho, eta;
if (not l in Integers) or (l <= 0) then
Error("level must be a positive integer."); fi;
if l = 1then
output := [];
Info(InfoSL2Reps, 1, "SL2Reps : At level 1, there is only the trivial irrep.");
return [
rec(
S := [[1]],
T := [[1]],
degree := 1,
level := 1,
name := "Xi_0"
)
]; else
factors := PrimePowersInt(l);
if Length(factors) = 2then # level is a prime power
Info(InfoSL2Reps, 1, "SL2Reps : Constructing irreps of level ", l, ", which is a prime power.");
return _SL2IrrepsPPLOfLevel(factors[1], factors[2]); else # tensor irreps of the prime power factors
Info(InfoSL2Reps, 1, "SL2Reps : Constructing irreps of prime-power level.");
pp_lists := []; for i in [1..(Length(factors) / 2)] do
Info(InfoSL2Reps, 1, "SL2Reps : Level ", (factors[2*i-1])^(factors[2*i]), ":");
pp_lists[i] := _SL2IrrepsPPLOfLevel(factors[2*i-1], factors[2*i]);
od;
# Possible alternate approach: make lists of the indices and take the Cartesian # thereof instead.
Info(InfoSL2Reps, 1, "SL2Reps : Constructing tensor products.");
pp_lists := Cartesian(pp_lists);
output := [];
for i in pp_lists do
rho := rec(
S := [[1]],
T := [[1]],
degree := 1,
level := 1,
name := "Xi_0"
);
for eta in i do
rho := rec(
S := KroneckerProduct(rho.S, eta.S),
T := KroneckerProduct(rho.T, eta.T),
degree := rho.degree * eta.degree,
level := rho.level * eta.level, # level is Lcm(level(rho), level(eta)), but they're coprime.
name := _SL2ConcatNames(rho.name,eta.name)
);
od;
Add(output, rho);
od;
# Sort by degree.
SortBy(output, x -> x.degree);
for i in [1 .. Length(output)] do
Info(InfoSL2Reps, 1, "SL2Reps : ", i, ": ( ", output[i].name, " ) [d: ", output[i].degree, ", l: ", output[i].level ,"]");
od;
Info(InfoSL2Reps, 1, "SL2Reps : Total count: ", Length(output));
return output; fi; fi; end );
# C_2 tensor R_4^2(r,3,chi), for chi != 1, r in {1,3}. # A = {+-1}, so there is a single non-trivial character, indexed by [0,1]. # NOTE: C_2 tensor R_4^2(1,1,[0,1]) is iso to R_4^2(1,3,nu)_1, and # C_2 tensor R_4^2(3,1,[0,1]) is iso to R_4^2(3,3,nu)_1. # Hence they are not exceptional. for r in [1,3] do
rho := SL2IrrepR(2,4,2,r,3,[0,1])[1];
rho := [-1 * rho[1], -1 * rho[2]];
name := Concatenation("Xi_6 tensor R_4^2(", String(r), ",3,[0,1])");
_SL2RecordIrrep(irrep_list, name, rho, 2^4);
od;
# N_3(chi)+ tensor R_4^0(1,7,psi)+, for psi(alpha) = -1, psi(-1) = 1, # chi primitive and chi^2 = 1. # Calculated via Kronecker product. # For the N part, chars. are [1,0] and [1,3] (see the ld=3 section earlier). # # First construct R_4^0(1,7,psi)+. Char is [1,1]. SL2IrrepR gives a list with + and then -.
x := SL2IrrepR(2, 4, 0, 1, 7, [1, 1]); for j in [0,3] do # Construct N_3(chi)+. SL2IrrepR again returns a list.
rho := SL2IrrepN(2, 3, [1, j]);
rho := [
KroneckerProduct(rho[1][1], x[1][1]),
KroneckerProduct(rho[1][2], x[1][2]), 12
];
name := Concatenation("N_3([1,", String(j), "])+ tensor R_4^0(1,7,[1,1])+");
_SL2RecordIrrep(irrep_list, name, rho, 2^4);
od;
# C_3 tensor R_5^2(r,1,chi)_1, for chi NOT primitive and r in {1,3}. # A = <alpha> x <-1>, so non-primitive characters are indexed by [0,0] and [0,1]. for r in [1,3] do for j in [0,1] do
rho := SL2IrrepR(2, 5, 2, r, 1, [0,j])[1];
rho := [E(4) * rho[1], -E(4) * rho[2], 12];
name := Concatenation("Xi_9 tensor R_5^2(", String(r), ",1,[0,", String(j), "])_1");
_SL2RecordIrrep(irrep_list, name, rho, 2^5);
od;
od;
# C_2 tensor R_6^4(r,t,nu)_1, for r in {1,3,5,7}, t in {1,3}. for r in [1,3,5,7] do for t in [1,3] do
rho := SL2IrrepR(2,6,4,r,t,[0,0])[1];
rho := [-1 * rho[1], -1 * rho[2]];
name := Concatenation("Xi_6 tensor R_6^4(", String(r), ",", String(t), ",nu)_1");
_SL2RecordIrrep(irrep_list, name, rho, 2^6);
od;
od;
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