# Tensor Categories Magma allows tensors and tensor spaces to change categories. Unless a user specifies otherwise, all tensors are assigned a category that is natural to the method by which it was created. For example a tensor created from an algebra will be assigned an algebra category, whereas a tensor created by structure constants will be assigned the Albert homotopism category [[Albert, 1942](../../references.md#cite-albert-fundamentals)]. Tensor categories influence the behavior of commands such as kernels and images as well as the algebraic invariants such as derivation algebras of a tensor. Our conventions follow [[Wilson, 2013](../../references.md#cite-wilson-division)]. In particular, given a tensor $T$ framed by $[U_{\nu},\dots,U_0]$ then a tensor category for $T$ will specify a function $A:[\nu]\to \{-1,0,1\}$ along with a partition ${\cal P}$ of $[\nu]$ such that the following rules apply to the tensors and morphisms in the category. **(1)** for each tensor $T$ framed by $[U_{\nu},\dots,U_0]$, if $X\in{\cal P}$, then $$ \forall i,j\in X,\quad U_i=U_j. $$ **(2)** Given a second tensor $S$ framed by $[V_v,\dots,V_0]$, a morphism $f:T\to S$ (Magma type `Hmtp`) will be a list $[f_{\nu},\dots,f_0]$ of homomorphisms as follows: **•** (Covariant) if $A(i)=1$ then $f_i:U_i\to V_i$; **•** (Constant) if $A(i)=0$ then $U_i=V_i$ and $f_i=1_{U_i}$; or else **•** (Contravariant) $A(i)=-1$ and $f_i:U_i\leftarrow V_i$. So if $A(0)=1$ then $$ \left\langle \sum_{i\in A^{-1}(1)} u_i f_i +\sum_{j\not\in A^{-1}(-1)} v_j\right\rangle_S = \left\langle \sum_{i\in A^{-1}(1)} u_i +\sum_{j\not\in A^{-1}(-1)} v_j f_j\right\rangle_T f_0; $$ if $A(0)=0$ then $$ \left\langle \sum_{i\in A^{-1}(1)} u_i f_i +\sum_{j\not\in A^{-1}(-1)} v_j\right\rangle_S = \left\langle \sum_{i\in A^{-1}(1)} u_i +\sum_{j\not\in A^{-1}(-1)} v_j f_j\right\rangle_T; $$ else $A(0)=-1$ and $$ \left\langle \sum_{i\in A^{-1}(1)} u_i f_i +\sum_{j\not\in A^{-1}(-1)} v_j\right\rangle_S f_0 = \left\langle \sum_{i\in A^{-1}(1)} u_i +\sum_{j\not\in A^{-1}(-1)} v_j f_j\right\rangle_T. $$ Magma manages internally the differences between vectors and covectors and more generally tensors and cotensors. Both types are issued the Magma type `TenSpcElt`. For operations sensitive to the difference, Magma stores a value of co/contra-variance of the tensor as a property of the tensor category. This is the third general property stored in Magma’s tensor category type `TenCat`. We use the phrase tensor category exclusively for categories that describe tensors and tensor spaces. In other words, the data structure of a tensor category is a function $A:[\nu]\rightarrow \{-1,0,1\}$ and a partition ${\cal P}$ of $[\nu]$. It is useful to distinguish from tensors and cotensors at the categorical level, so a tensor category is either covariant or contravariant as well (in the latter case, referred to as a cotensor category). ## Constructing Tensor Categories ### `TensorCategory(A, P): [RngIntElt], {SetEnum} -> TenCat` ### `TensorCategory(A, P): Map, {SetEnum} -> TenCat` Sets up a covariant tensor space category with specified direction of arrows $A$, and a partition ${\cal P}$ indicating variables to be treated as equivalent. The fiber $A^{-1}(1)$ denotes the covariant variables, $A^{-1}(0)$ identifies the constant variables, and $A^{-1}(-1)$ marks the contra-variant variables. ### `CotensorCategory(A, P): [RngIntElt], {SetEnum} -> TenCat` ### `CotensorCategory(A, P): Map, {SetEnum} -> TenCat` Sets up a contra-variant tensor space category with specified direction of arrows $A$, and a partition ${\cal P}$ indicating variables to be treated as equivalent. The fiber $A^{-1}(1)$ denotes the covariant variables, $A^{-1}(0)$ identifies the constant variables, and $A^{-1}(-1)$ marks the contra-variant variables. ### `Example: Basic Cat Const (ex-2873fe)` We demonstrate the basic tensor category constructor. The difference between `TensorCategory` and `CotensorCategory` is only that the former is covariant and the latter is contravariant. ```magma > C := TensorCategory([1,0,-1], {{0},{1},{2}}); > C; Tensor category of valence 3 (->,==,<-) ({ 1 },{ 2 },{ 0 }) > IsCovariant(C); true > > arrows := map< {1..5} -> {1} | x :-> 1 >; > C := CotensorCategory(arrows, {{1..5}}); > C; Cotensor category of valence 6 (->,->,->,->,->,==) ({ 0 },{ 1 .. 5 }) > IsContravariant(C); true ``` ### `HomotopismCategory(v : parameters): RngIntElt -> TenCat` ```magma Contravariant: BoolElt Default: false ``` Returns Albert’s homotopism category – all modules categories are covariant and no duplicates considered. Set the optional parameter `Contravariant` to `true` to make it a cotensor category. ### `CohomotopismCategory(v): RngIntElt -> TenCat` Returns the cohomotopism category – all domain modules categories are covariant, the codomain is contravariant, and no duplicates considered. ### `AdjointCategory(v, s, t): RngIntElt, RngIntElt, RngIntElt -> TenCat` ### `LinearCategory(v, s, t): RngIntElt, RngIntElt, RngIntElt -> TenCat` Returns the tensor category where all modules are constant except in position $s$ and $t$. Both $s$ and $t$ are in $[v]$. Position $s$ is covariant, position $t$ is contravariant. ### `Example: Ten Cat Special (ex-f54041)` Now we look at a few special tensor category constructors. The default tensor category is the homotopism category, so we construct the homotopism category using `TensorCategory` and verify they are equivalent. ```magma > C := TensorCategory([1,1,1,1], {{i} : i in [0..3]}); > C; Tensor category of valence 4 (->,->,->,->) ({ 1 },{ 2 },{ 0 },{ 3 }) > HomotopismCategory(4) eq C; true ``` The other special tensor categories can be constructed using `TensorCategory` as well, but we just construct a few to show their properties. ```magma > CohomotopismCategory(3); Tensor category of valence 3 (->,->,<-) ({ 1 },{ 2 },{ 0 }) > > AdjointCategory(5, 4, 1); Tensor category of valence 5 (<-,==,==,->,==) ({ 1 },{ 0, 2, 3 },{ 4 }) ``` ## Operations on Tensor Categories In this section the basic operations for tensor categories are described. ### `C1 eq C2: TenCat, TenCat -> BoolElt` Returns `true` if the tensor categories are the same. ### `Valence(C): TenCat -> RngIntElt` Returns the valence of the tensor category. ### `Arrows(C): TenCat -> SeqEnum` Returns the sequence of arrows of the tensor category. A $-1$ signifies a contravariant index, a $0$ signifies a constant index, and a $1$ signifies a covariant index. ### `RepeatPartition(C): TenCat -> SetEnum` Returns the repeat partition for the tensor category. ### `IsCovariant(C): TenCat -> BoolElt` ### `IsContravariant(C): TenCat -> BoolElt` Returns `true` if the tensor category is covariant or contravariant. ### `Example: Ten Cat Properties (ex-d61054)` We obtain basic properties of tensor categories. ```magma > C := CotensorCategory([1,0,-1,1],{{4,3},{1},{2}}); > C; Cotensor category of valence 5 (->,==,<-,->,==) ({ 1 },{ 2 },{ 0 },{ 3, 4 }) > > Valence(C); 5 > Arrows(C); [ 1, 0, -1, 1 ] > IsContravariant(C); true > RepeatPartition(C); { { 1 }, { 2 }, { 3, 4 } } ``` ## Categorical Operations In this section, we define subtensors, local ideals, ideals, and quotients of tensors. For the following definitions fix a tensor $t\in T$, with frame $\oslash_{a\in[\nu]}U_a$—that is $U_{\nu}\times\cdots\times U_1\rightarrowtail U_0$. A tensor $s:V_{\nu}\times \cdots\times V_1\rightarrowtail V_0$ is a *subtensor* of $t$ if for all $a$, $V_a\leq U_a$. For $A\subseteq [\nu]-0$, a tensor $s:V_{\nu}\times \cdots\times V_1\rightarrowtail V_0$ is an $A$*-local ideal* of $t$ if $s$ is a subtensor of $t$ and for each $a\in A$, $$ \langle s \,|\, V_{\nu},\dots, V_{a+1}, U_a, V_{a-1},\dots, V_1\rangle \leq V_0. $$ A tensor $s$ is an *ideal* of $t$ if it is a $\{1,\dots,\nu\}$-local ideal of $t$. The $A$*-local quotient* of a tensor $t$ by an $A$-local ideal $s$ is the tensor $q:U_{\nu}/V_{\nu} \times \cdots \times U_1/V_1\rightarrowtail U_0/V_0$ where for all $| \overline{\nu}\rangle$, $$ \langle q \,|\, \overline{\nu}\rangle \equiv \langle q \,|\, \nu\rangle \rm{mod} V_0. $$ The *quotient* of a tensor $t$ by an ideal $s$ is the $\{1,\dots,\nu\}$-local quotient of $t$ by $s$. ### Categorical Operations on Tensors We include functions defined for the category of tensors. Most functions are currently defined only for the homotopism category. #### `Subtensor(T, S): TenSpcElt, List -> TenSpcElt` #### `Subtensor(T, S): TenSpcElt, SeqEnum -> TenSpcElt` Returns the smallest submap of $T$ containing $S$. #### `Subtensor(T, D, C): TenSpcElt, List, Any -> TenSpcElt` #### `Subtensor(T, D, C): TenSpcElt, SeqEnum, Any -> TenSpcElt` Returns the smallest submap of $T$ containing $D$ in the domain and $C$ in the codomain. #### `IsSubtensor(T, S): TenSpcElt, TenSpcElt -> BoolElt` Decides whether $S$ is a subtensor of $T$. #### `Example: Subtensors (ex-b3fa65)` We construct the tensor $t$ given by octonion multiplication. The quaternions $H$ are a subalgebra of $A= O$ generated by the first four basis elements. However, $H$ cannot be coerced into $A$ because of how Magma organizes algebras. ```magma > A := OctonionAlgebra(Rationals(), -1, -1, -1); > t := Tensor(A); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 8 over Rational Field U1 : Full Vector space of degree 8 over Rational Field U0 : Full Vector space of degree 8 over Rational Field > H := sub< A | A.1, A.2, A.3, A.4 >; > H; Algebra of dimension 4 with base ring Rational Field ``` There are multiple ways to get the subtensor of multiplication from $H$. We will create $H\times H\rightarrowtail H$ as a subtensor of $A\times A\rightarrowtail A$. ```magma > H_gens := [A.i : i in [1..4]]; > s := Subtensor(t, [*H_gens, H_gens, A!0*]); > s; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 8, dimension 4 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) Echelonized basis: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) U1 : Vector space of degree 8, dimension 4 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) Echelonized basis: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) U0 : Vector space of degree 8, dimension 4 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) Echelonized basis: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) ``` Note that because $H$ cannot be coerced into $A$ (i.e. `A!H` produces an error), a subtensor of `AlgGen` cannot be done by `Subtensor(t, [H, H, H])`. Now we will construct the tensor straight from $H$. There is a subtle difference between the subtensor from $A$ and the tensor from $H$—namely, the frame is different. ```magma > s2 := Tensor(H); > s2; 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 > s eq s2; false > Eltseq(s) eq Eltseq(s2); true ``` #### `LocalIdeal(T, S, I): TenSpcElt, List, {RngIntElt} -> TenSpcElt` #### `LocalIdeal(T, S, I): TenSpcElt, SeqEnum, {RngIntElt} -> TenSpcElt` Returns the local ideal of $T$ at $I$ containing $S$. #### `LocalIdeal(T, D, C, I): TenSpcElt, List, Any, {RngIntElt} -> TenSpcElt` #### `LocalIdeal(T, D, C, I): TenSpcElt, SeqEnum, Any, {RngIntElt} -> TenSpcElt` Returns the local ideal of $T$ at $I$ containing $D$ in the domain and $C$ in the codomain. #### `LocalIdeal(T, S, I): TenSpcElt, TenSpcElt, {RngIntElt} -> TenSpcElt` Returns the local ideal of $T$ at $I$ containing $S$ as a submap. #### `IsLocalIdeal(T, S, I): TenSpcElt, TenSpcElt, {RngIntElt} -> BoolElt` Decides if $S$ is a local ideal of $T$ at $I$. #### `Example: Local Ideals (ex-96befc)` We use the same tensor $t$ as the previous example, multiplication in $A= O$, and we construct the subtensor $t_2$ given by multiplication in $H$. We construct a subtensor $s$ of $t$ as the submap containing $\langle A_2\rangle \times \langle A_1, A_4\rangle \rightarrowtail \langle 0\rangle$, which is equal to $\langle A_2\rangle \times \langle A_1, A_4\rangle \rightarrowtail \langle A_2A_1, A_2A_4\rangle$. ```magma > A := OctonionAlgebra(Rationals(), -1, -1, -1); > t := Tensor(A); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 8 over Rational Field U1 : Full Vector space of degree 8 over Rational Field U0 : Full Vector space of degree 8 over Rational Field > H_gens := [A.i : i in [1..4]]; > t2 := Subtensor(t, [*H_gens, H_gens, H_gens*]); > s := Subtensor(t, [* A.2, [A.1, A.4], A!0 *]); > s; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 8, dimension 1 over Rational Field Generators: (0 1 0 0 0 0 0 0) Echelonized basis: (0 1 0 0 0 0 0 0) U1 : Vector space of degree 8, dimension 2 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 0 0 1 0 0 0 0) Echelonized basis: (1 0 0 0 0 0 0 0) (0 0 0 1 0 0 0 0) U0 : Vector space of degree 8, dimension 2 over Rational Field Generators: (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) Echelonized basis: (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) ``` The quaternions are a subalgebra of $A$ generated by $\{A_1,A_2,A_3,A_4\}$. Therefore, the $\{2\}$-local ideal of $s$ in $t$ must contain $A$ in the codomain. However, the $\{2\}$-local ideal of $s$ in $t_2$ must only contain $\langle A_1, A_2, A_3, A_4\}$ in the codomain. ```magma > s1 := LocalIdeal(t, s, {2}); > Codomain(s1); Full Vector space of degree 8 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) (0 0 0 0 1 0 0 0) (0 0 0 0 0 1 0 0) (0 0 0 0 0 0 1 0) (0 0 0 0 0 0 0 1) > s2 := LocalIdeal(t2, s, {2}); > Codomain(s2); Vector space of degree 8, dimension 4 over Rational Field Generators: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) Echelonized basis: (1 0 0 0 0 0 0 0) (0 1 0 0 0 0 0 0) (0 0 1 0 0 0 0 0) (0 0 0 1 0 0 0 0) ``` #### `Ideal(T, S): TenSpcElt, List -> TenSpcElt` #### `Ideal(T, S): TenSpcElt, SeqEnum -> TenSpcElt` Returns the ideal of $T$ containing $S$. #### `Ideal(T, D, C): TenSpcElt, List, Any -> TenSpcElt` #### `Ideal(T, D, C): TenSpcElt, SeqEnum, Any -> TenSpcElt` Returns the ideal of $T$ containing $D$ in the domain and $C$ in the codomain. #### `Ideal(T, S): TenSpcElt, TenSpcElt -> TenSpcElt` Returns the ideal of $T$ containing $S$ as a submap. #### `IsIdeal(T, S): TenSpcElt, TenSpcElt -> BoolElt` Decides if $S$ is an ideal of $T$. #### `Example: Ideals (ex-fad333)` First we will construct the tensor from the ${\mathbb{Q}}$-algebra, ${\mathbb{Q}}^5$. ```magma > T := KTensorSpace(Rationals(), [5,5,5]); > A := VectorSpace(Rationals(), 5); > t := T!0; > for i in [1..5] do > Assign(~t, [i,i,i], 1); > end for; > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 5 over Rational Field U1 : Full Vector space of degree 5 over Rational Field U0 : Full Vector space of degree 5 over Rational Field > SystemOfForms(t); [ [1 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0], [0 0 0 0 0] [0 1 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0], [0 0 0 0 0] [0 0 0 0 0] [0 0 1 0 0] [0 0 0 0 0] [0 0 0 0 0], [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 1 0] [0 0 0 0 0], [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 1] ] ``` Now we will construct the ideal tensor from the subtensor containing $\langle A_1\rangle \times \langle A_2\rangle \rightarrowtail \langle A_3\rangle$. Note that the $\{2\}$-local ideal must include $\langle A_2,A_3\rangle$ in the codomain, and the $\{1\}$-local ideal must contain $\langle A_1,A_3\rangle$ in the codomain. ```magma > s := Ideal(t, [A.1, A.2, A.3]); > s; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 5, dimension 1 over Rational Field Generators: (1 0 0 0 0) Echelonized basis: (1 0 0 0 0) U1 : Vector space of degree 5, dimension 1 over Rational Field Generators: (0 1 0 0 0) Echelonized basis: (0 1 0 0 0) U0 : Vector space of degree 5, dimension 3 over Rational Field Generators: (1 0 0 0 0) (0 1 0 0 0) (0 0 1 0 0) Echelonized basis: (1 0 0 0 0) (0 1 0 0 0) (0 0 1 0 0) ``` Finally, we verify that the subtensor containing $\langle A_1\rangle \times \langle A_2\rangle \rightarrowtail \langle A_2, A_3\rangle$ is not an ideal. ```magma > r := Subtensor(t, [A.1, A.2], [A.2, A.3]); > IsIdeal(t, r); false ``` #### `LocalQuotient(T, S, I : parameters): TenSpcElt, TenSpcElt, {RngIntElt} -> TenSpcElt, Hmtp` ```magma Check: BoolElt Default: true ``` Returns the local quotient of $T$ by $S$ at $I\subseteq[\nu]-0$. If you know $S$ is a local ideal of $T$ at $I$, set `Check` to `false` to skip the verification. A homotopism is also returned, mapping from $T$ to $T/S$. #### `Quotient(T, S : parameters): TenSpcElt, TenSpcElt -> TenSpcElt, Hmtp` #### `T / S: TenSpcElt, TenSpcElt -> TenSpcElt, Hmtp` ```magma Check: BoolElt Default: true ``` Returns the quotient of $T$ by $S$. If you know $S$ is an ideal of $T$, set `Check` to `false` to skip the verification. A homotopism is also returned, mapping from $T$ to $T/S$. #### `Example: Quotients (ex-0f6850)` We will demonstrate one of the most common uses for quotienting tensors: constructing the associated fully nondegenerate tensor. We first construct a tensor with a nontrivial radical given by matrix multiplication: $\rm{Mat}_{3\times 2}({\mathbb{Q}})\times {\mathbb{Q}}^3\rightarrowtail {\mathbb{Q}}^3$, where we take a projection of ${\mathbb{Q}}^3$ onto ${\mathbb{Q}}^2$ in the 1 coordinate. ```magma > K := Rationals(); > F := [*KMatrixSpace(K, 3, 2), VectorSpace(K, 3), VectorSpace(K, 3)*]; > mult := function(x) > return Transpose(x[1]*Matrix(2, 1, Eltseq(x[2])[2..3])); > end function; > t := Tensor(F, mult); > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 6 over Rational Field U1 : Full Vector space of degree 3 over Rational Field U0 : Full Vector space of degree 3 over Rational Field > s := Subtensor(t, [*[F[1].1, F[1].4], F[2], F[3]*]); > s; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 6, dimension 2 over Rational Field Generators: (1 0 0 0 0 0) (0 0 0 1 0 0) Echelonized basis: (1 0 0 0 0 0) (0 0 0 1 0 0) U1 : Full Vector space of degree 3 over Rational Field Generators: (1 0 0) (0 1 0) (0 0 1) U0 : Full Vector space of degree 3 over Rational Field Generators: (1 0 0) (0 1 0) (0 0 1) > IsFullyNondegenerate(s); false ``` Now we will construct the ideal $r$ that evaluates to 0 and the largest subspace of ${\mathbb{Q}}^3$ not contained in the image. ```magma > r := Ideal(t, [*F[1]!0, F[2].1, F[3].3*]); > r; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 6, dimension 0 over Rational Field Generators: U1 : Vector space of degree 3, dimension 1 over Rational Field Generators: (1 0 0) Echelonized basis: (1 0 0) U0 : Vector space of degree 3, dimension 1 over Rational Field Generators: (0 0 1) Echelonized basis: (0 0 1) > IsIdeal(t, r); true ``` Finally, we quotient $s$ by $r$ to obtain a fully nondegenerate tensor. ```magma > q := s/r; > q; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 2 over Rational Field > IsFullyNondegenerate(q); true ``` ### Categorical Operations on Tensor Spaces We have categorical notions for tensor spaces as well, and these are inherited from the module structure on tensor spaces. #### `SubConstructor(T, L): TenSpc, Any -> TenSpc, Map` #### `sub< T | L >: TenSpc, Any -> TenSpc, Map` Returns the subtensor space of $T$ generated by the tensors in the sequence $L$. #### `IsSubtensorSpace(T, S): TenSpc, TenSpc -> BoolElt` Decides if the tensor space $S$ is a subtensor space of $T$. #### `Example: Subtensor Spaces (ex-40cd43)` We will construct the subspace $S$ of symmetric forms from the tensor space $T$ with frame ${\mathbb{Q}}^2\times{\mathbb{Q}}^2\rightarrowtail {\mathbb{Q}}$. ```magma > K := Rationals(); > T := KTensorSpace(K, [2,2,1]); > T; Tensor space of dimension 4 over Rational Field with valence 3 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > S := sub< T | T.1, T.2+T.3, T.4 >; > S; Tensor space of dimension 3 over Rational Field with valence 3 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > IsSymmetric(S); true ``` Now we will construct the subspace $A$ of alternating forms from $T$. ```magma > A := sub< T | T.2-T.3 >; > A; Tensor space of dimension 1 over Rational Field with valence 3 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > IsAlternating(A); true ``` Now we verify that $A$ is not a subtensor space of $S$, and thus, we have constructed a direct decomposition of $T$ into its symmetric and alternating space. ```magma > IsSubtensorSpace(S, A); false ``` #### `QuoConstructor(T, X): TenSpc, Any -> TenSpc, Map` #### `quo< T | X >: TenSpc, Any -> TenSpc, Map` #### `T / S: TenSpc, TenSpc -> TenSpc, Map` Returns the quotient tensor space of $T$ by $S$. #### `Example: Quotient Tensor Spaces (ex-87c8de)` We pick up with the same tensor spaces as the previous example: $T$ has frame ${\mathbb{Q}}^2\times{\mathbb{Q}}^2\rightarrowtail {\mathbb{Q}}$, $S$ is the symmetric subspace, and $A$ is the alternating subspace. ```magma > K := Rationals(); > T := KTensorSpace(K, [2,2,1]); > T; Tensor space of dimension 4 over Rational Field with valence 3 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > S := sub< T | T.1, T.2+T.3, T.4 >; > A := sub< T | T.2-T.3 >; ``` Now we construct the quotient of $T$ by $A$. The result is not a symmetric tensor space. Note that $Q_2$ is equivalent to a symmetric tensor modulo $A$, but this choice is arbitrary. ```magma > Q := T/A; > Q; Tensor space of dimension 3 over Rational Field with valence 3 U2 : Full Vector space of degree 2 over Rational Field U1 : Full Vector space of degree 2 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > SystemOfForms(Q.1); [ [1 0] [0 0] ] > SystemOfForms(Q.2); [ [0 0] [1 0] ] > SystemOfForms(Q.3); [ [0 0] [0 1] ] ``` ## Homotopisms Magma provides functions for homotopisms, i.e. morphisms of tensors. Homotopisms are also equipped with a tensor category. Because homotopisms can contain multiple maps between modules, it is not really clear what is meant by the domain or codomain of a given homotopism. In this context, the *domain* of a homotopism $H$ will refer to the tensor, $t:U_{\nu}\times \cdots \times U_1\rightarrowtail U_0$, where an arrow equal to $1$ at coordinate $a$ will mean that the corresponding map at coordinate $a$ has domain equal to $U_a$. And in the same vain, the tensor $s:V_{\nu} \times\cdots \times V_1\rightarrowtail V_0$ is the *codomain* of $H$ if an arrow equal to $1$ at coordinate $a$ implies that the corresponding map at coordinate $a$ has codomain equal to $V_a$. ### Constructions of Homotopisms #### `Homotopism(T, S, M : parameters): TenSpcElt, TenSpcElt, List -> Hmtp` #### `Homotopism(T, S, M : parameters): TenSpcElt, TenSpcElt, SeqEnum -> Hmtp` #### `Homotopism(T, S, M, C : parameters): TenSpcElt, TenSpcElt, List, TenCat -> Hmtp` #### `Homotopism(T, S, M, C : parameters): TenSpcElt, TenSpcElt, SeqEnum, TenCat -> Hmtp` ```magma Check: BoolElt Default: true ``` ```magma Check: BoolElt Default: true ``` Returns the homotopism from $T$ to $S$ given by the list of maps $M$ and the category $C$. The default tensor category is the same as tensor categories for $T$ and $S$. If the maps $M$ will produce a homotopism, then set `Check` to `false` to skip the verification. #### `Homotopism(M, C): List, TenCat -> Hmtp` #### `Homotopism(M, C): SeqEnum, TenCat -> Hmtp` Returns the homotopism given by the maps in $M$ with tensor category $C$. #### `IsHomotopism(T, s, H): TenSpcElt, TenSpcElt, Hmtp -> BoolElt` #### `IsHomotopism(T, s, M): TenSpcElt, TenSpcElt, List -> BoolElt` #### `IsHomotopism(T, s, M): TenSpcElt, TenSpcElt, SeqEnum -> BoolElt` #### `IsHomotopism(T, s, M, C): TenSpcElt, TenSpcElt, List, TenCat -> BoolElt` #### `IsHomotopism(T, s, M, C): TenSpcElt, TenSpcElt, SeqEnum, TenCat -> BoolElt` Decides if the list of maps $M$ induces a homotopism from $T$ to $S$ in the tensor category $C$. The default tensor category is the homotopism category. If it does induce a homotopism, it is also returned. #### `Example: Homotopism Const (ex-6fd856)` We will construct two symmetric tensors $t,s: {\bf F}_{3}^3 \times {\bf F}_{3}^3 \rightarrowtail {\bf F}_{3}^3$ and apply permutations to the bases. ```magma > T := KTensorSpace(GF(3), [3,3,3]); > t := T.1+T.14+T.27; > SystemOfForms(t); [ [1 0 0] [0 0 0] [0 0 0], [0 0 0] [0 1 0] [0 0 0], [0 0 0] [0 0 0] [0 0 1] ] > s := (T.4+T.10)+(T.8+T.20)+(T.18+T.24); > SystemOfForms(s); [ [0 1 0] [1 0 0] [0 0 0], [0 0 1] [0 0 0] [1 0 0], [0 0 0] [0 0 1] [0 1 0] ] ``` Now we construct a homotopism from $t$ to $t$ given by apply a permutation matrix in every coordinate. ```magma > P := PermutationMatrix(GF(3), [2,1,3]); > H := Homotopism(t, t, [*P, P, P*]); > H; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: [0 1 0] [1 0 0] [0 0 1] U1 -> V1: [0 1 0] [1 0 0] [0 0 1] U0 -> V0: [0 1 0] [1 0 0] [0 0 1] ``` Note that this permutation matrix does not induce a homotopism of $s$. ```magma > IsHomotopism(s, s, [*P, P, P*]); false ``` #### `Example: Mixed Homotopisms (ex-d02f12)` Homotopisms can take mixed categories of maps. To reuse the above example, we can encode the permutation as a `Map` and construct homotopisms from these types. ```magma > V := VectorSpace(GF(3), 3); > T := TensorSpace([V, V, V]); > t := T.1+T.14+T.27; > P := PermutationMatrix(GF(3), [2,1,3]); > f := hom< V -> V | [, , ] >; > H := Homotopism(t, t, [*f, f, f*]); > H; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: Mapping from: Full Vector space of degree 3 over GF(3) to Full Vector space of degree 3 over GF(3) U1 -> V1: Mapping from: Full Vector space of degree 3 over GF(3) to Full Vector space of degree 3 over GF(3) U0 -> V0: Mapping from: Full Vector space of degree 3 over GF(3) to Full Vector space of degree 3 over GF(3) ``` Furthermore, we can input lists with types `Mtrx` and `Map` included. ```magma > H2 := Homotopism(t, t, [*P, f, P*]); > H2; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: [0 1 0] [1 0 0] [0 0 1] U1 -> V1: Mapping from: Full Vector space of degree 3 over GF(3) to Full Vector space of degree 3 over GF(3) U0 -> V0: [0 1 0] [1 0 0] [0 0 1] ``` ### Basic Operations with Homotopisms We provide some operations for homotopisms. #### `H1 * H2: Hmtp, Hmtp -> Hmtp` Returns the composition of the homotopisms $H_1$ and $H_2$. #### `H . a: Hmtp, RngIntElt -> Map` Returns the map on the $a$th coordinate. #### `Example: Homotopism Ops (ex-c4051c)` We construct a nondegenerate alternating form $t$ on $V={\mathbb{Q}}^6$. The group of isometries are isomorphic to $\rm{Sp}(6, {\mathbb{Q}})$, the group generated by all transvections. We construct a transvection $L$ and a corresponding matrix. ```magma > V := VectorSpace(Rationals(), 6); > T := KTensorSpace(Rationals(), [6, 6, 1]); > t := T.2-T.7+T.16-T.21+T.30-T.35; > t; Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 6 over Rational Field U1 : Full Vector space of degree 6 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > > u := V.2+2*V.3-V.5; > L := map< V -> V | x :-> x + (x*t*u)[1]*u >; > L; Mapping from: ModTupFld: V to ModTupFld: V given by a rule [no inverse] > M := Matrix(6, 6, [V.i @ L : i in [1..6]]); > M; [ 1 1 2 0 -1 0] [ 0 1 0 0 0 0] [ 0 0 1 0 0 0] [ 0 -2 -4 1 2 0] [ 0 0 0 0 1 0] [ 0 1 2 0 -1 1] ``` We construct a homotopism from the transvection. ```magma > H := Homotopism(t, t, [*L, L, IdentityMatrix(Rationals(), 1)*]); > H; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] U1 -> V1: Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] U0 -> V0: [1] ``` Since $H$ is an isometry of $t$, $H^2$ is also an isometry of $t$. We verify that the 2-coordinate map of $H^2$ is exactly $M^2$. ```magma > H2 := H*H; > H2; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field Composition of Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] and Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] U1 -> V1: Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field Composition of Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] and Mapping from: Full Vector space of degree 6 over Rational Field to Full Vector space of degree 6 over Rational Field given by a rule [no inverse] U0 -> V0: [1] > M2 := Matrix(6, 6, [V.i @ H2.2 : i in [1..6]]); > M2; [ 1 2 4 0 -2 0] [ 0 1 0 0 0 0] [ 0 0 1 0 0 0] [ 0 -4 -8 1 4 0] [ 0 0 0 0 1 0] [ 0 2 4 0 -2 1] > M^2 eq M2; true ``` #### `Precompose(T, f, a): TenSpcElt, Map, RngIntElt -> TenSpcElt` #### `Precompose(T, M, a): TenSpcElt, Mtrx, RngIntElt -> TenSpcElt` If $a>0$, then the tensor returned is the tensor that has been pre-composed by the map $f$ or matrix $M$. #### `T @ H: TenSpcElt, Hmtp -> TenSpcElt` If $H$ is a cohomotopism (a homotopism in the cohomotopism category), then either the domain or codomain of $H$ is returned, depending on the orientation of the arrows of $H$. ### Basic Properties of Homotopisms #### `Domain(H): Hmtp -> TenSpcElt` Returns the domain tensor of $H$. #### `Codomain(H): Hmtp -> TenSpcElt` Returns the codomain tensor of $H$. #### `Maps(H): Hmtp -> List` Returns the list of maps for the various modules in the domain and codomain tensors. #### `TensorCategory(H): Hmtp -> TenCat` Returns the tensor category of $H$. #### `ChangeTensorCategory(H, C): Hmtp, TenCat -> Hmtp` #### `ChangeTensorCategory(~H, C): Hmtp, TenCat` Changes the tensor category of $H$ to the given category. #### `Valence(H): Hmtp -> RngIntElt` Returns the valence of the underlying tensor category of the homotopism $H$. #### `Kernel(H): Hmtp -> TenSpcElt` Returns the kernel of $H$ as an ideal of its domain tensor. #### `Image(H): Hmtp -> TenSpcElt` Returns the image of $H$ as a submap of the codomain tensor. #### `Example: Homotopism Props (ex-5a1efc)` We demonstrate how to access properties of a homotopism. We construct tensors $t:{\mathbb{Q}}^4\times{\mathbb{Q}}^4\rightarrowtail {\mathbb{Q}}$ and $s:{\mathbb{Q}}^6\times{\mathbb{Q}}^6\rightarrowtail{\mathbb{Q}}$ given by the dot product. ```magma > t := Tensor(IdentityMatrix(Rationals(), 4), 2, 1); > s := Tensor(IdentityMatrix(Rationals(), 6), 2, 1); > Z := ZeroMatrix(Rationals(), 4, 6); > M := InsertBlock(Z, IdentityMatrix(Rationals(), 4), 1, 1); > H := Homotopism(t, s, [*M, M, IdentityMatrix(Rationals(), 1)*]); > H; Maps from U2 x U1 >-> U0 to V2 x V1 >-> V0. U2 -> V2: [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0] U1 -> V1: [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0] U0 -> V0: [1] ``` Like with maps, we can obtain standard properties of homotopisms. ```magma > Domain(H); 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 1 over Rational Field > Codomain(H); Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space of degree 6 over Rational Field U1 : Full Vector space of degree 6 over Rational Field U0 : Full Vector space of degree 1 over Rational Field > Maps(H); [* [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0], [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0], [1] *] > TensorCategory(H); Tensor category of valence 3 (->,->,->) ({ 1 },{ 2 },{ 0 }) ``` When the image and kernel can be computed for the each of the maps in the homotopism, then the image and kernel can be computed for the homotopism. ```magma > Im := Image(H); > Im; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 6, dimension 4 over Rational Field Echelonized basis: (1 0 0 0 0 0) (0 1 0 0 0 0) (0 0 1 0 0 0) (0 0 0 1 0 0) U1 : Vector space of degree 6, dimension 4 over Rational Field Echelonized basis: (1 0 0 0 0 0) (0 1 0 0 0 0) (0 0 1 0 0 0) (0 0 0 1 0 0) U0 : Full Vector space of degree 1 over Rational Field > Ker := Kernel(H); > Ker; Tensor of valence 3, U2 x U1 >-> U0 U2 : Vector space of degree 4, dimension 0 over Rational Field U1 : Vector space of degree 4, dimension 0 over Rational Field U0 : Vector space of degree 1, dimension 0 over Rational Field ``` #### `Shuffle(H, g): Hmtp, GrpPermElt -> Hmtp` #### `Shuffle(H, g): Hmtp, [RngIntElt] -> Hmtp` Just like the shuffle for tensors, this returns the shuffle of the homotopism $H$. This is a functor from one tensor category to another and changes the order of the maps to $$ \{H_{\nu^g},\dots, H_{1^g}, H_{0^g}\}. $$ In order to be defined, $g\in{\operatorname{Sym}}(\{0,\dots,\nu \})$. If $0^g\ne 0$, then both the image and pre-image of $0$ under $g$ will be replaced by their $K$-dual space. For cotensors, $g\in{\operatorname{Sym}}(\{1,\dots,\nu\})$. Sequences $[a_1,\dots,a_{\nu+1}]$ will be interpreted as a permutation in one-line notation.