Linear Invariants of Tensors
Invariants for Bilinear Tensors
The following intrinsics are specialized for tensors of valence 3, but equivalent intrinsics for general tensors are presented in the proceeding subsection.
- AdjointAlgebra(B): TenSpcElt -> AlgMat
Returns the adjoint \(*\)-algebra of the given Hermitian bilinear map \(B\), represented on \(\rm{End}(U_2)\). This is using algorithms from StarAlge, see the
AdjointAlgebraintrinsic in [BW]. If the current version of StarAlge is not attached, the default Magma version will be used instead.
- Example: Adjoint Algebra of a p-Group Tensor (ex-55dbd9)
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Given the context of [BW12b], we construct a tensor from a \(p\)-group and compute its adjoint algebra.
> G := SmallGroup(3^7, 7000); > t := pCentralTensor(G); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 4 over GF(3) U1 : Full Vector space of degree 4 over GF(3) U0 : Full Vector space of degree 3 over GF(3)
Unlike other intrinsics that compute invariants of tensors,
AdjointAlgebraexploits the fact that \(t\) is Hermitian so that the adjoint algebra is faithfully represented on \(\rm{End}(U_2)=\rm{End}(U_1)\).> A := AdjointAlgebra(t); > A; Matrix Algebra of degree 4 and dimension 4 with 4 generators over GF(3) > A.1; [1 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 1]
Because \(A\) is constructed from algorithms for \(*\)-algebras, we can apply other algorithms from that package specifically dealing with the involution on \(A\).
> RecognizeStarAlgebra(A); true > SimpleParameters(A); [ <"symplectic", 2, 3> ] > Star(A); Mapping from: AlgMat: A to AlgMat: A given by a rule [no inverse]
- LeftNucleus(B : parameters): TenSpcElt -> AlgMat
op: BoolElt Default: false
Returns the left nucleus of the bilinear map \(B\) as a subalgebra of \(\rm{End}_K(U_2)\times \rm{End}_K(U_0)\). In previous versions of TensorSpace (and eMAGma), the left nucleus was returned as a subalgebra of \(\rm{End}(U_2)^\circ\times \rm{End}(U_0)^\circ\). To enable this, set the optional argument
optotrue.
- MidNucleus(B): TenSpcElt -> AlgMat
Returns the mid nucleus of the bilinear map \(B\) as a subalgebra of \(\rm{End}_K(U_2)\times \rm{End}_K(U_1)\).
- RightNucleus(B): TenSpcElt -> AlgMat
Returns the right nucleus of the bilinear map \(B\) as a subalgebra of \(\rm{End}_K(U_1)\times \rm{End}_K(U_0)\).
- Example: Nuclei Embed in the Derivation Algebra (ex-75db37)
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We will verify a theorem from [FMW19] and [Wil17]: all the nuclei of a tensor embed into the derivation algebra. We construct the tensor given by \((3\times 4\times 5)\)-matrix multiplication.
> K := Rationals(); > A := KMatrixSpace(K, 3, 4); > B := KMatrixSpace(K, 4, 5); > C := KMatrixSpace(K, 3, 5); > F := func< x | x[1]*x[2] >; > t := Tensor([A, B, C], F); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 12 over Rational Field U1 : Full Vector space of degree 20 over Rational Field U0 : Full Vector space of degree 15 over Rational Field
Because matrix multiplication is associative, the left, middle, and right nuclei contain \(\rm{Mat}_3({\mathbb{Q}})\), \(\rm{Mat}_4({\mathbb{Q}})\), and \(\rm{Mat}_5({\mathbb{Q}})\) respectively. In fact, the following computation shows that this is equality.
> L := LeftNucleus(t : op := true); > M := MidNucleus(t); > R := RightNucleus(t); > Dimension(L), Dimension(M), Dimension(R); 9 16 25 > D := DerivationAlgebra(t); > Dimension(D); 49
Now we will embed these nuclei into the derivation algebra of \(t\).
> Omega := KMatrixSpace(K, 47, 47); > Z1 := ZeroMatrix(K, 20, 20); > L_L2, L2 := Induce(L, 2); > L_L0, L0 := Induce(L, 0); > embedL := map< L -> Omega | x :-> > DiagonalJoin(<Transpose(x @ L_L2), Z1, Transpose(x @ L_L0)>) >; > > Z0 := ZeroMatrix(K, 15, 15); > M_M2, M2 := Induce(M, 2); > M_M1, M1 := Induce(M, 1); > embedM := map< M -> Omega | x :-> > DiagonalJoin(<x @ M_M2, -Transpose(x @ M_M1), Z0>) >; > > Z2 := ZeroMatrix(K, 12, 12); > R_R1, R1 := Induce(R, 1); > R_R0, R0 := Induce(R, 0); > embedR := map< R -> Omega | x :-> > DiagonalJoin(<Z2, x @ R_R1, x @ R_R0>) >; > > Random(Basis(L)) @ embedL in D; true > Random(Basis(M)) @ embedM in D; true > Random(Basis(R)) @ embedR in D; true
Invariants of General Multilinear Maps
The following functions can be used for general tensors.
- Centroid(T): TenSpcElt -> AlgMat
- Centroid(T, A): TenSpcElt, {RngIntElt} -> AlgMat
Returns the \(A\)-centroid of the tensor as a subalgebra of
\[\prod_{a\in A}\rm{End}(U_a),\]where \(A\subseteq [\nu]\). If no \(A\) is given, it is assumed that \(A=[\nu]\). If \(t\) is contained in a category where coordinates \(a\) and \(b\) (also contained in \(A\)) are fused together, then the corresponding operators on those coordinates will be equal.
- Example: Centroid of a Multiplication Tensor (ex-71158e)
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The centroid \(C\) of a tensor \(t\) is the largest ring for which \(t\) is \(C\)-linear, see [FMW19, Theorem D]. To demonstrate this, we will construct the tensor given by multiplication of the splitting field of \(f(x)=x^4-x^2-2\) over \({\mathbb{Q}}\). However, this field won’t explicitly be given with the tensor data.
> A := MatrixAlgebra(Rationals(), 4); > R<x> := PolynomialRing(Rationals()); > F := sub< A | A!1, CompanionMatrix(x^4-x^2-2) >; > F; Matrix Algebra of degree 4 with 2 generators over Rational Field > t := Tensor(F); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 4 over Rational Field U1 : Full Vector space of degree 4 over Rational Field U0 : Full Vector space of degree 4 over Rational Field
The centroid is the field \({\mathbb{Q}}(\sqrt{2},i)\).
> C := Centroid(t); > C; Matrix Algebra of degree 12 with 4 generators over Rational Field > sub< C | C.1 > eq C; true > forall{ c : c in Generators(C) | IsInvertible(c) }; true > IsCommutative(C); true > MinimalPolynomial(C.1); x^4 + 2*x^2 - 8 > Factorization(MinimalPolynomial(C.1)); [ <x^2 - 2, 1>, <x^2 + 4, 1> ]
- DerivationAlgebra(T): TenSpcElt -> AlgMatLie
- DerivationAlgebra(T, A): TenSpcElt, {RngIntElt} -> AlgMatLie
Returns the \(A\)-derivation Lie algebra of the tensor as a Lie subalgebra of
\[\prod_{a\in A}\rm{End}(U_a),\]where \(A\subseteq [\nu]\). If no \(A\) is given, it is assumed that \(A=[\nu]\). If \(t\) is contained in a category where coordinates \(a\) and \(b\) (also contained in \(A\)) are fused together, then the corresponding operators on those coordinates will be equal.
- Nucleus(T, a, b): TenSpcElt, RngIntElt, RngIntElt -> AlgMat
- Nucleus(T, A): TenSpcElt, SetEnum -> AlgMat
Returns the \(A\)-nucleus, for \(A=\{a,b\}\) (\(a\ne b\)), of the tensor as a subalgebra of \(\rm{End}(U_i)\times \rm{End}(U_j)\), where \(i=\max(a,b)\) and \(j=\min(a,b)\). If \(j>0\), then replace \(\rm{End}(U_j)\) with \(\rm{End}(U_j)^\circ\). If \(T\) is contained in a category where coordinates \(a\) and \(b\) are fused together, then the corresponding operators on those coordinates will not be forced to be equal.
- Example: Restricting the Derivation Algebra (ex-84e946)
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In a previous example, we embedded the nuclei of a tensor into the derivation algebra. For a tensor \(t: U_{\nu}\times \cdots\times U_1\rightarrowtail U_0\), the derivation algebra is represented in
\[\Omega=\prod_{a\in[\nu]}\rm{GL}(U_a).\]We will restrict the derivation algebra to
\[\prod_{c\notin\{a,b\}} \rm{GL}(U_c)\]for distinct \(a,b\in[\nu]\). From [FMW19, Lemma 4.11], the kernel of this restriction is equal to Nuc\(_{\{a,b\}}(t)^-\). We will just verify that the dimensions match.
We will construct a tensor given by matrix multiplication:
\[\rm{Mat}_{3\times 4}({\bf F}_{2})\times \rm{Mat}_{4\times 2}({\bf F}_{2}) \times \rm{Mat}_{2\times 2}({\bf F}_{2}) \rightarrowtail \rm{Mat}_{3\times 2}({\bf F}_{2}).\]> A := KMatrixSpace(GF(2), 3, 4); > B := KMatrixSpace(GF(2), 4, 2); > C := KMatrixSpace(GF(2), 2, 2); > D := KMatrixSpace(GF(2), 3, 2); > trip := func< x | x[1]*x[2]*x[3] >; > t := Tensor([A, B, C, D], trip); > t; Tensor of valence 4, U3 x U2 x U1 >-> U0 U3 : Full Vector space of degree 12 over GF(2) U2 : Full Vector space of degree 8 over GF(2) U1 : Full Vector space of degree 4 over GF(2) U0 : Full Vector space of degree 6 over GF(2)
Now we will compute the derivation algebra of
t. We choose \(a=3\) and \(b=2\), so the \(\{3,2\}\)-nucleus is \(\rm{Mat}_{4\times 4}({\bf F}_{2})\).> D := DerivationAlgebra(t); > Dimension(D); 32 > N32 := Nucleus(t, 3, 2); > N32; Matrix Algebra of degree 20 with 16 generators over GF(2)
To construct the restriction of \(\rm{Der}(t)\) into \(\rm{GL}(U_1)\times\rm{GL}(U_0)\) we will use the
Inducefunction.> Omega_10 := KMatrixSpace(GF(2), 10, 10); > D_vs := sub< KMatrixSpace(GF(2), 30, 30) | Basis(D) >; > pi1, D1 := Induce(D, 1); > pi0, D0 := Induce(D, 0); > res := hom< D_vs -> Omega_10 | > [<x, DiagonalJoin(x @ pi1, x @ pi0)> : x in Basis(D)] >; > res; Mapping from: ModMatFld: D_vs to ModMatFld: Omega_10 > Kernel(res); KMatrixSpace of 30 by 30 matrices and dimension 16 over GF(2)
- SelfAdjointAlgebra(t, a, b): TenSpcElt, RngIntElt, RngIntElt -> ModMatFld
Returns the self-adjoint elements of the \(ab\)-nucleus of \(t\) as a subspace of \(\rm{End}(U_a)\), with \(a,b\in[\nu]\) and \(a\ne b\). It is not required that \(t\) be in a tensor category with coordinates \(a\) and \(b\) fused. Unlike other invariants associated to tensors, the self-adjoint algebra is not currently stored with the tensor.
- TensorOverCentroid(T): TenSpcElt -> TenSpcElt, Hmtp
If the given tensor \(T\) is framed by \(K\)-vector spaces, then the returned tensor is framed by \(E\)-vector spaces where \(E\) is the residue field of the centroid. The returned homotopism is an isotopism of the \(K\)-tensors. This only works if the centroid of \(T\) is a finite commutative local ring. We employ the algorithms developed by Brooksbank and Wilson [BW15] to efficiently determine if a matrix algebra is cyclic.
- Example: Recovering a Field with the Centroid (ex-93b5e7)
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In the context of groups, centroids can be used to recover an underlying field of a matrix group, even if the given group is not input as such. Here we will construct the exponent-\(p\) central tensor of the Sylow 2-subgroup of \(\rm{GL}(3,\rm{GF}(2^{10}))\). We will not print the
GrpPCversion of this group as the number of relations is very large.> U := ClassicalSylow(GL(3, 2^10), 2); > U.3; [ 1 $.1^2 0] [ 0 1 0] [ 0 0 1] > G := PCPresentation(UnipotentMatrixGroup(U)); > #G eq 2^30; true > t := pCentralTensor(G); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 20 over GF(2) U1 : Full Vector space of degree 20 over GF(2) U0 : Full Vector space of degree 10 over GF(2)
Even though our tensor right now is \({\bf F}_{2}^{20}\times {\bf F}_{2}^{20}\rightarrowtail{\bf F}_{2}^{10}\), we know it is the 2-dimensional alternating form over \(\rm{GF}(2^{10})\). We will construct the centroid, and then rewrite our tensor over the centroid to get the tensor we expect.
> C := Centroid(t); > C; Matrix Algebra of degree 50 and dimension 10 with 1 generator over GF(2) > IsCyclic(C) and IsSimple(C); true > s := TensorOverCentroid(t); > s; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 2 over GF(2^10) U1 : Full Vector space of degree 2 over GF(2^10) U0 : Full Vector space of degree 1 over GF(2^10)