# 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 `AdjointAlgebra` intrinsic in [[Brooksbank and Wilson, n.d.](../../references.md#cite-bw-staralge)]. If the current version of StarAlge is not attached, the default Magma version will be used instead. ### `Example: Adjoint Alge (ex-55dbd9)` Given the context of [[Brooksbank and Wilson, 2012](../../references.md#cite-bw-isometry)], we construct a tensor from a $p$-group and compute its adjoint algebra. ```magma > 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, `AdjointAlgebra` exploits the fact that $t$ is Hermitian so that the adjoint algebra is faithfully represented on $\rm{End}(U_2)=\rm{End}(U_1)$. ```magma > 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 $A$. ```magma > 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` ```magma 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 `op` to `true`. ### `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: Going Nuclear (ex-75db37)` We will verify a theorem from [[First *et al.*, 2019](../../references.md#cite-fmw-densors)] and [[Wilson, 2017](../../references.md#cite-wilson-lmr)]: all the nuclei of a tensor embed into the derivation algebra. We construct the tensor given by $(3\times 4\times 5)$-matrix multiplication. ```magma > 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. ```magma > 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 $t$. ```magma > 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() >; > > Z0 := ZeroMatrix(K, 15, 15); > M_M2, M2 := Induce(M, 2); > M_M1, M1 := Induce(M, 1); > embedM := map< M -> Omega | x :-> > DiagonalJoin() >; > > Z2 := ZeroMatrix(K, 12, 12); > R_R1, R1 := Induce(R, 1); > R_R0, R0 := Induce(R, 0); > embedR := map< R -> Omega | x :-> > DiagonalJoin() >; > > 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 (ex-71158e)` The centroid $C$ of a tensor $t$ is the largest ring for which $t$ is $C$-linear, see [[First *et al.*, 2019](../../references.md#cite-fmw-densors), 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. ```magma > A := MatrixAlgebra(Rationals(), 4); > R := 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 ${\mathbb{Q}}(\sqrt{2},i)$. ```magma > 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)); [ , ] ``` ### `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: Restrict Derivation (ex-84e946)` 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 [[First *et al.*, 2019](../../references.md#cite-fmw-densors), 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: $$ \rm{Mat}_{3\times 4}({\bf F}_{2})\times \rm{Mat}_{4\times 2}({\bf F}_{2}) \times \rm{Mat}_{2\times 2}({\bf F}_{2}) \rightarrowtail \rm{Mat}_{3\times 2}({\bf F}_{2}). $$ ```magma > 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 $\rm{Mat}_{4\times 4}({\bf F}_{2})$. ```magma > 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 `Induce` function. ```magma > 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 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 [[Brooksbank and Wilson, 2015](../../references.md#cite-bw-module-iso)] to efficiently determine if a matrix algebra is cyclic. ### `Example: Centroid Unipotent (ex-93b5e7)` 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 `GrpPC` version of this group as the number of relations is very large. ```magma > 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. ```magma > 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) ```