| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/sl2reps/gap/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 30.10.2022 mit Größe 83 kB |
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Quelle IrrepLists.gi Sprache: unbekannt |
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Spracherkennung für: .gi vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen] #
# 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 := []; if 1 = 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 > 3 then # 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) / 2 then 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) = 1 then 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) / 2 then 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 = 1 then # 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) / 2 then 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), " _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), " _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) / 2 then 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), " _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; od; od; # p = 2, ld = 1. l := 2; if 1 = degree then # Xi_6 = C_2, with s = t = -1. rho := [ [[-1]], [[-1]] ]; name := "Xi_6"; _SL2RecordIrrep(irrep_list, name, rho, l); fi; if 2 = 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; if 3 = 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; if 2 = 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; if 1 = 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; if 6 = 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); # R_3^0(1,3,nu)_1 and C_3 tensor R_3^0(1,3,nu)_1. # Matrix given in notes S. 1.6.3. rho := [ -1/2 * [ [ 0, 0, 1, 1, 1, 1], [ 0, 0, -1, -1, 1, 1], [ 1, -1, 0, 0, 1, -1], [ 1, -1, 0, 0, -1, 1], [ 1, 1, 1, -1, 0, 0], [ 1, 1, -1, 1, 0, 0], ], DiagonalMat([1, -1, E(8), E(8)^5, E(8)^3, E(8)^7]) ]; name := "R_3^0(1,3,nu)_1"; _SL2RecordIrrep(irrep_list, name, rho, 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; if 4 = 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; if 2 = 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; if 3 = 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; if 24 = 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; if 8 = 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; if 6 = 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); rho := SL2IrrepR(2, 4, 0, 1, t, [1,1]); name := Concatenation("R_4^0(1,", String(t), ",[1,1])+"); _SL2RecordIrrep(irrep_list, name, rho[1], l); name := Concatenation("R_4^0(1,", String(t), ",[1,1])-"); _SL2RecordIrrep(irrep_list, name, rho[2], l); od; # 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); rho := SL2IrrepR(2,4,2,r,3,[0,1])[1]; name := Concatenation("R_4^2(", String(r), ",3,[0,1])"); _SL2RecordIrrep(irrep_list, name, rho, l); 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, l); od; # 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; if 3 = 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; if 12 = 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; if 48 = 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; if 16 = 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; if 12 = 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; if 24 = 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; if 6 = 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; if 3*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; if 2^(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; if 3*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( _SL2RecordIrrep(irrep_list, name, rho, l); od; od; od; fi; if 3*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( _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( _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) _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) _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) _SL2RecordIrrep(irrep_list, name, rho, l); od; od; od; od; fi; if 3*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), " _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) _SL2RecordIrrep(irrep_list, name, rho, l); rho := SL2IrrepR(2, ld, ld-2, r, t, [1,1])[1]; name := Concatenation("R_", String(ld), "^", String(ld-2), "(", String(r), ",", String(t) _SL2RecordIrrep(irrep_list, name, rho, l); od; od; if ld = 6 then # 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) _SL2RecordIrrep(irrep_list, name, rho, l); rho := SL2IrrepR(2,ld,ld-3,r,t,[2,0])[1]; name := Concatenation("R_", String(ld), "^", String(ld-3), "(", String(r), ",", String(t) _SL2RecordIrrep(irrep_list, name, rho, l); od; od; fi; fi; od; 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 = 0 then # The trivial irrep Xi_0 = C_1. _SL2RecordIrrep(irrep_list, "Xi_0", [[[1]], [[1]]], 1); elif p > 2 then if ld = 1 then # 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); # N_1(nu), the Steinberg representation. rho := SL2IrrepN(p, 1, [0, 0])[1]; name := "N_1(nu)"; _SL2RecordIrrep(irrep_list, name, rho, l); else # ld >= 2. # 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) / 2 then 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) = 1 then 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) / 2 then 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 = 1 then # 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) / 2 then 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), " _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), " _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) / 2 then 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), " _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 = 1 then # 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 = 2 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); # 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 = 3 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); # R_3^0(1,3,nu)_1 and C_3 tensor R_3^0(1,3,nu)_1. # Matrix given in notes S. 1.6.3. rho := [ -1/2 * [ [ 0, 0, 1, 1, 1, 1], [ 0, 0, -1, -1, 1, 1], [ 1, -1, 0, 0, 1, -1], [ 1, -1, 0, 0, -1, 1], [ 1, 1, 1, -1, 0, 0], [ 1, 1, -1, 1, 0, 0], ], DiagonalMat([1, -1, E(8), E(8)^5, E(8)^3, E(8)^7]) ]; name := "R_3^0(1,3,nu)_1"; _SL2RecordIrrep(irrep_list, name, rho, 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); # 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); # 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 = 4 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); # 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); rho := SL2IrrepR(2, 4, 0, 1, t, [1,1]); name := Concatenation("R_4^0(1,", String(t), ",[1,1])+"); _SL2RecordIrrep(irrep_list, name, rho[1], l); name := Concatenation("R_4^0(1,", String(t), ",[1,1])-"); _SL2RecordIrrep(irrep_list, name, rho[2], l); od; # 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); rho := SL2IrrepR(2,4,2,r,3,[0,1])[1]; name := Concatenation("R_4^2(", String(r), ",3,[0,1])"); _SL2RecordIrrep(irrep_list, name, rho, l); 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, l); od; # 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); 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; # 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 = 5 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; # 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); 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; # 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( _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( _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( _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) _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) _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) _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), " _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) _SL2RecordIrrep(irrep_list, name, rho, l); rho := SL2IrrepR(2, ld, ld-2, r, t, [1,1])[1]; name := Concatenation("R_", String(ld), "^", String(ld-2), "(", String(r), ",", String(t) _SL2RecordIrrep(irrep_list, name, rho, l); od; od; if ld = 6 then # 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) _SL2RecordIrrep(irrep_list, name, rho, l); rho := SL2IrrepR(2,ld,ld-3,r,t,[2,0])[1]; name := Concatenation("R_", String(ld), "^", String(ld-3), "(", String(r), ",", String(t) _SL2RecordIrrep(irrep_list, name, rho, l); od; od; fi; fi; fi; 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] := []; _SL2RecordIrrep(prime_power_reps[1], "Xi_0", [[[1]], [[1]]], 1); # Xi_0 = C_1 _SL2RecordIrrep(prime_power_reps[1], "Xi_6", [[[-1]], [[-1]]], 2); # Xi_6 = C_2 _SL2RecordIrrep(prime_power_reps[1], "Xi_4", [[[1]], [[E(3)]]], 3); # Xi_4 = R_1(1)- with _SL2RecordIrrep(prime_power_reps[1], "Xi_8", [[[1]], [[E(3)^2]]], 3); # Xi_8 = R_1(2)- wi _SL2RecordIrrep(prime_power_reps[1], "Xi_3", [[[-E(4)]], [[E(4)]]], 4); # Xi_3 = C_4 _SL2RecordIrrep(prime_power_reps[1], "Xi_9", [[[E(4)]], [[-E(4)]]], 4); # Xi_9 = C_3 _SL2RecordIrrep(prime_power_reps[1], "Xi_2", [[[-1]], [[-E(3)^2]]], 6); # = Xi_6 * Xi_8 _SL2RecordIrrep(prime_power_reps[1], "Xi_10", [[[-1]], [[-E(3)]]], 6); # = Xi_6 * Xi_4 _SL2RecordIrrep(prime_power_reps[1], "Xi_1", [[[E(4)]], [[E(12)]]], 12); # = Xi_9 * Xi_4 _SL2RecordIrrep(prime_power_reps[1], "Xi_5", [[[E(4)]], [[E(12)^5]]], 12); # = Xi_9 * Xi_ _SL2RecordIrrep(prime_power_reps[1], "Xi_7", [[[-E(4)]], [[E(12)^7]]], 12); # = Xi_3 * Xi _SL2RecordIrrep(prime_power_reps[1], "Xi_11", [[[-E(4)]], [[E(12)^11]]], 12); # = Xi_3 * Info(InfoSL2Reps, 1, "SL2Reps : 12 irreps of degree 1 found."); for i in DivisorsInt(degree) do if i = 1 then continue; fi; Info(InfoSL2Reps, 1, "SL2Reps : Degree ", i, ":"); prime_power_reps[i] := _SL2IrrepsPPLOfDegree(i); od; Info(InfoSL2Reps, 1, "SL2Reps : Constructing tensor products."); ConstructIrreps := function(rho, factors, start) local i, eta, output, name, new_start; if Length(factors) = 0 then # 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) = 1 then 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: 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 = 1 then 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) = 2 then # 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: od; Info(InfoSL2Reps, 1, "SL2Reps : Total count: ", Length(output)); return output; fi; fi; end ); InstallGlobalFunction( SL2IrrepsExceptional, function() local irrep_list, rho, name, w, x, j, r, t; irrep_list := []; # C_2 tensor R_2^0(1,3)_1. # 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]) ]; rho := [-1*rho[1], -1*rho[2]]; name := "Xi_6 tensor R_2^0(1,3)_1"; _SL2RecordIrrep(irrep_list, name, rho, 2^2); # C_3 tensor R_3^0(1,3,nu)_1. # Matrix given in notes S. 1.6.3. rho := [ -1/2 * [ [ 0, 0, 1, 1, 1, 1], [ 0, 0, -1, -1, 1, 1], [ 1, -1, 0, 0, 1, -1], [ 1, -1, 0, 0, -1, 1], [ 1, 1, 1, -1, 0, 0], [ 1, 1, -1, 1, 0, 0], ], DiagonalMat([1, -1, E(8), E(8)^5, E(8)^3, E(8)^7]) ]; 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, 2^3); # 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; Info(InfoSL2Reps, 1, "SL2Reps : ", Length(irrep_list), " exceptional irreps found."); return irrep_list; end ); [Dauer der Verarbeitung: 0.54 Sekunden, vorverarbeitet 2026-04-26] |
2026-05-26