Chris Liu
Math gradudate student at Colorado State University
Algebraic Invariants of Tensors: Algorithms and Decompositions
Chris Liu
Dissertation Defense
May 8, 2026
Plan
- Tensors and their algebraic invariants
- First result: a faster algorithm to compute the adjoint and derivation algebra of tensors
- Second result: theorem on the algebraic invariants on the product of tensors
Tensors and their algebraic invariants
\( T \) is \( (a \times b \times c) \) grid of numbers
Tensors are multiway grids of numbers with a multilinear interpretation
As a bilinear map
\[ \tau: \mathbb{R}^a \times \mathbb{R}^b \rightarrowtail \mathbb{R}^c \]
As a trilinear form
\[ t: \mathbb{R}^a \times \mathbb{R}^b \times \mathbb{R}^c \rightarrowtail \mathbb{R}\]
\[t \left(\sum_i u_i e_i, \sum_j u_je_j, \sum_k u_ke_k \right) = \sum_{ijk} T_{ijk} u_iv_jw_k\]
Example 1: Multiplication
Let \( A \) be a \(\mathbb{F}\)-vector space with bilinear multiplication \( \mu: A \times A \rightarrowtail A \) (\(\mathbb{F}\)-algebra)
For ordered basis \( (e_1,\ldots, e_n )\) of \(A\), coordinatize \(\mu\) by a \( (n \times n \times n) \) grid of numbers \(T\) satisfying \( \mu(e_i,e_j) = \sum_k T_{ijk}e_k \).
For \(A = \frac{\mathbb{F}[x]}{x^2-3x+2} \)
Change of bases
Example 2: Discord chat logs
A cube of numbers, where \( (i,j,k) \) is the number of times word \(i\) was said by user \(j\) at time \(k\) in a Discord chatroom
\( \text{Users} = \{ \text{Andrea}, \text{Bob}, \text{Chris}, \text{Dave}, \text{Eve}, \ldots \} \)
\( \text{Days of the week} = \{\text{Monday}, \text{Tuesday}, \ldots \} \)
\( \text{Words} = \{\text{apple}, \text{break}, \text{colorado},\text{dog}, \ldots \} \)
\( \text{Groups} = \{ \alpha_1 \cdot \text{Chris} + \alpha_2 \cdot \text{Dave} + \alpha_3 \cdot \text{Eve}, \ldots \} \)
\( \text{Time clusters} = \{\beta_1 \cdot \text{Saturday} + \beta_2 \cdot \text{Sunday}, \ldots \} \)
\( \text{Themes} = \{ \gamma_1\cdot \text{hockey} + \gamma_2 \cdot \text{puck} + \gamma_3 \cdot \text{goalie}, \ldots \} \)
Linear combinations
Data mining algorithm
Algebraic invariants detect structure
Source: Optimal Search Spaces for Tensor Problems, Figure 1.1
Adjoints detect
Given \( t: U \times V \rightarrowtail W \)
$$\operatorname{Adj}(t) = \{(X,Y) \mid (\forall u,v) \;\; t(Xu,v) = t(u,Yv) \}$$
Analogy: Given a bilinear form \( \langle \cdot \mid \cdot \rangle : V \times V \rightarrowtail \mathbb{F} \) and \(A \in \operatorname{End}(V)\), its adjoint \( A^{\ast} \) satisfies \( \langle Au \mid v \rangle = \langle u \mid A^{\ast} v \rangle \)
(transpose for the dot product)
Myasnikov '90, Wilson '08 and others use the algebra, i.e
Theorem
Exists \(\mathcal{E} = \{e_1,\ldots, e_n \} \subset \text{Adj}(t) \),
\( \sum_i e_i = 1\), and \(e_ie_j = e_i \) if \( i = j \), otherwise \(0\)
if and only if
Exists \(\perp\)-decomposition, \(U := \bigoplus_i U_i\) and \(V := \bigoplus_i V_i \),
\[ t(U_i, V_j) = t(U_i,V_i) \quad \text{if } i = j, \text{otherwise } 0 \]
Derivations detect
Theorem (Brooksbank, Kassabov, Wilson '24)
For \( (u,v,w) \) eigenvectors of \( (X,Y,Z) \) in \( \text{Der}(t) \) with eigenvalues \( (\kappa , \lambda, \rho) \),
\[w^{\dagger}Z(t(u,v)) = w^{\dagger}(t(Xu,v) + t(u,Yv))\]
\[ \rho w^{\dagger}(t(u,v))-w^{\dagger}(\kappa t(u,v) - \lambda t(u, v)) = 0\]
By distributive property,
\[ (\rho - \kappa - \lambda) (w^{\dagger}(t(u,v))) = 0 \]
Source: Detecting cluster patterns in tensor data, Figure 3
Given \( t: U \times V \rightarrowtail W \)
$$\operatorname{Der}(t) = \{(X,Y,Z) \mid (\forall u,v) \; Z(t(u,v)) = t(Xu,v) + t(u,Yv) \}$$
Analogy: The product rule in Calculus to understand multiplication. The derivative \( \frac{d}{dx}(fg) = (\frac{d}{dx}f)g + f(\frac{d}{dx}g) \)
The surface \(z=xy\)
Eigenbasis gives cluster patterns
Tensors vs tensor product of vector spaces?
Given \(U,V\) vector spaces, the tensor product space is \( (U \otimes V, \varphi: U \times V \rightarrowtail U \otimes V) \)
such that for all bilinear maps \(f: U \times V \rightarrowtail W\), there exists unique induced linear map \(\tilde{f}: U \otimes V \rightarrow W\) such that \(f(u,v) = \tilde{f}(\varphi(u,v))\).
- \(U \otimes V\) is spanned by \( \{ \varphi(u,v) : u \in U, v \in V \}\)
- Pure tensors are elements \( \varphi(u,v) \in U \otimes V \), often denoted as \(u \otimes v\)
- Elements of \(U \otimes V\) are tensors in our sense after identifying \( U \otimes V \) with \((U \otimes V)^{\ast} \) for instance by a basis
"Theorem" (L.)
Faster algorithms to compute the adjoint and derivation algebra of tensors
Three step algorithm
- solving a smaller restricted subsystem
- lifting the subsystem solution to a global candidate solution
- verify the global candidate solution
Let \(A \in \mathbb{F}^{a \times b \times c} \)
\( (R,S,T) = (A,-A, 0) \) gives adjoint
(Joint with James B. Wilson, Joshua Maglione)
R \in \mathbb{F}^{r \times b \times c}\\
S \in \mathbb{F}^{a \times s \times c}\\
T \in \mathbb{F}^{a \times b \times c}
X \in \mathbb{F}^{a \times r}\\
Y \in \mathbb{F}^{s \times b}
Given
Find
Such that
(\forall i \in [c]) \; XR_i + S_iY = T_i
Simultaneous Sylvester System
Recall given \( t: \mathbb{F}^{a} \times \mathbb{F}^{b} \rightarrowtail \mathbb{F}^{c} \) $$\operatorname{Adj}(t) = \{(X,Y) \mid (\forall u,v)\;\; t(Xu,v) = t(u,Yv)\}$$
Corresponding linear system
Example
R_1 =
\begin{bmatrix}
1 & 5 \\
0 & -1 \\
2 & 11
\end{bmatrix},
\quad R_2 =
\begin{bmatrix}
0 & 1 \\
1 & 1 \\
-1 & 1
\end{bmatrix}
S_1 =
\begin{bmatrix}
1 & 0 & 3 & 0\\
1 & 1 & 10 & -1
\end{bmatrix},
\quad S_2 =
\begin{bmatrix}
0 & 1 & 7 & -1\\
1 & -1 & -4 & 1
\end{bmatrix}
T_1 =
\begin{bmatrix}
1 & 19\\
-11 & 4
\end{bmatrix},
\quad T_2 =
\begin{bmatrix}
-10 & 4\\
8 & 2
\end{bmatrix}
\begin{array}{|ccc|ccc|cccc|cccc||c|}
\hline
X_{11} & X_{12} & X_{13} & X_{21} & X_{22} & X_{23}
& Y_{11} & Y_{21} & Y_{31} & Y_{41} & Y_{12} & Y_{22} & Y_{32} & Y_{42} & T_{ijk} \\
\hline
1 & 0 & 2 & & & & 1 & 0 & 3 & 0 & & & & & 1 \\
0 & 1 & -1 & & & & 0 & 1 & 7 & -1 & & & & & -10 \\
\hline
5 & -1 & 11 & & & & & & & & 1 & 0 & 3 & 0 & 19 \\
1 & 1 & 1 & & & & & & & & 0 & 1 & 7 & -1 & 4 \\
\hline
& & & 1 & 0 & 2 & 1 & 1 & 10 & -1 & & & & & -11 \\
& & & 0 & 1 & -1 & 1 & -1 & -4 & 1 & & & & & 8 \\
\hline
& & & 5 & -1 & 11 & & & & & 1 & 1 & 10 & -1 & 4 \\
& & & 1 & 1 & 1 & & & & & 1 & -1 & -4 & 1 & 2\\
\hline
\end{array}
(LMW) Idea: Avoid big linear system altogether with a solve-lift-check approach
Linear system given by matrix of size \( (abc) \times (ar + bs) \) is \(O(n^7) \) to solve
(complexity statements made with respect to \(n = a+b+c+r+s \) )
\text{Recall } R \in \mathbb{F}^{r \times b \times c}, S \in \mathbb{F}^{a \times s \times c}, T \in \mathbb{F}^{a \times b \times t}
Permute rows & columns
Idea: solve-lift-check approach
\( I \)
\( J \)
\( M_R \)
\( M_S \)
(L.) Distill to
- Solve \( Mx=b \) systems
- Solve \( MX=N(\theta) \) parametrized systems
- Solve \(Au+\alpha = Bv + \beta\) systems
- Restrict to \(I \subset [a]\) for \(S\), and \(J \subset [b] \) for \(R\)
- Solve restricted system for \( (X_{DR}, Y_{DR}) \)
- Backsub \( M_R X = N_R(Y_{DR}) \)
- Backsub \( M_S Y = N_S(X_{DR}) \)
- Verify
Given \(M \in \mathbb{F}^{m \times n} \), and \( N: \Theta \rightarrow \mathbb{F}^{m \times \ell} \) affine map, solve for \(X \in \mathbb{F}^{n \times \ell} \)
Given matrices \(A,B\) and constants \( \alpha, \beta\), solve for all pairs \( (u,v) \)
regular family of instances
- (Bounded dimension) \( \operatorname{dim}(\Theta_{DR}) \leq C\)
- (Lift feasibility if restricted system has solutions)
- \( N_R(Y_{DR}) \subseteq \operatorname{Im}(M_R) \)
- \( N_S(X_{DR}) \subseteq \operatorname{Im}(M_S) \)
- (Lift uniqueness) \( \operatorname{Ker}(M_R) = \operatorname{Ker}(M_S) = \{0\} \)
Theorem A (L.)
Given a regular family, and a black-box algorithm for \( (I,J) \), the solve-and-lift approach takes \(O(n^4)\) arithmetic operations for deterministic solutions.
Remark - The black-box restriction algorithm is to prove determinstic runtimes. \( \exists \) randomized algorithms.
\( I \)
\( J \)
\( M_R \)
\( M_S \)
Recall backsub \( M_R X = N_R(Y_{DR}) \), \( M_S Y = N_S(X_{DR}) \)
Assumption is reasonable for natural classes of instances (e.g generic cubic tensors with overdetermined restricted system)
Derivation Equation
Let \(A \in \mathbb{F}^{a \times b \times c} \)
\( (R,S,T) = (A,A, A) \) gives derivation
Recall given \( t: \mathbb{F}^{a} \times \mathbb{F}^{b} \rightarrowtail \mathbb{F}^c \)
$$\operatorname{Der}(t) = \{(X,Y,Z) \mid (\forall u,v) \; Z(t(u,v)) = t(Xu,v) + t(u,Yv) \}$$
R \in \mathbb{F}^{r \times b \times c}\\
S \in \mathbb{F}^{a \times s \times c}\\
T \in \mathbb{F}^{a \times b \times t}
X \in \mathbb{F}^{a \times r}\\
Y \in \mathbb{F}^{s \times b}\\
Z \in \mathbb{F}^{t \times c}
Given
Find
Such that
(\forall i \in [c]) \; XR_i + S_iY = \sum_j T_jZ_{ji}
Theorem B (L.)
Given a regular family, and a black-box algorithm for \( (I,J,K) \), the solve-and-lift approach takes \(O(n^{4.5})\) arithmetic operations for deterministic solutions.
Analogous derivation regular family ensure the bounded dimension, and unique lifting property
- Restrict to \(I \subset [a]\) for \(S\), \(J \subset [b] \) for \(R\), and \(K \subset [c] \) for \(T\)
- Solve restricted-system for \( (X_{TR}, Y_{TR}, Z_{TR}) \)
- Backsub for \( X,Y,Z \)
- Verify on unrestricted indices
Remark - Extra \( n^{0.5} \) factor is due to the triple restricted system needing to solve a linear system of \( O(n^{1.5})\) variables
t = [t_1, t_2], \text{ with} \\
t_1 = \begin{bmatrix}
10 & 5\\
5 & 6
\end{bmatrix} \quad
t_2 = \begin{bmatrix}
9 & 3\\ 3 & 7
\end{bmatrix}
\begin{array}{|cccc|cccc|cccc|}
\hline
x_{11}&x_{12}&x_{21}&x_{22}
&y_{11}&y_{21}&y_{12}&y_{22}
&z_{11}&z_{12}&z_{21}&z_{22} \\\hline
10& 5& & & 10& 5& & & -10& & -9& \\
9& 3& & & 9& 3& & & & -10& & -9\\
5& 6& & & & &10& 5 & -5& & -3& \\
3& 7& & & & & 9& 3 & & -5& & -3\\
& &10& 5 & 5& 6& & 0 & -5& & -3& \\
& & 9& 3 & 3& 7& & 0 & & -5& & -3\\
& & 5& 6 & & & 5& 6 & -6& & -7& \\
& & 3& 7 & & & 3& 7 & & -6& & -7\\
\hline
\end{array}
\( \text{Der}(t) \) is nullspace of
Derivation system performance
Sylvester system performance
Extras cut for time
Remark - As \(O(n^4)\) cost is needed to determinstically verify a solution, this is the best complexity for a determinstic solution we can expect.
"Theorem" (L.)
Decomposing the algebraic invariants of the product of tensors
Tensor products of algebras
Let \( A, B \) be unital associative \( \mathbb{F} \)-algebras
\( A \otimes B \) is a unital associative \(\mathbb{F}\)-algebra, with multiplication
\[ (a \otimes b)(c \otimes d) = ac \otimes bd \]
Example: Kronecker product of matricies
Example: Extending scalars (Recall \( \mathbb{H} \) are the Quaternions)
\( u = i_{\mathbb{H}} \otimes i_{\mathbb{C}} \) satisfies \( u^2 = 1 \), so the idempotent \(\frac{1+u}{2} \) splits the algebra
\mathbb{H} \otimes \mathbb{C} \cong \mathbb{M}_2(\mathbb{C})
\[ 1 \otimes 1 \mapsto \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, i \otimes 1 \mapsto \begin{bmatrix}i & 0\\ 0 & -i\end{bmatrix}, j \otimes 1 \mapsto \begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}, k \otimes 1 \mapsto \begin{bmatrix}0 & i \\ i & 0\end{bmatrix}\]
Define on multiplication tables
\begin{array}{|c|cc|}
\hline
& 1 & x \\ \hline
1 & 1 & x \\
x & x & -1 \\
\hline
\end{array}
\otimes
\begin{array}{|c|cc|}
\hline
& 1 & y \\ \hline
1 & 1 & y \\
y & y & -5y-10 \\
\hline
\end{array} =
\begin{array}{|c|cccc|}
\hline
& 1 & x & y & xy \\ \hline
1 & 1 & x & y & xy \\
x & x & -1 & xy & -y \\
y & y & xy & -5y-10 & -5xy-10x \\
xy & xy & -y & -5xy-10x & 5y+10 \\
\hline
\end{array}
\(\frac{\mathbb{F}[y]}{y^2+5y+10} \)
\( \mathbb{F}[\sqrt{-1}] \)
Let \( s: U \times V \rightarrowtail W \) and \(t: X \times Y \rightarrowtail Z \)
Define \( s \otimes t: (U \otimes X) \times (V \otimes Y) \rightarrowtail (W \otimes Z) \) as
\[ (s \otimes t)(u\otimes x, v \otimes y) = s(u,v) \otimes t(x,y) \]
Product of bilinear maps
Let \(s: \mathbb{F}^2 \times \mathbb{F} \rightarrowtail \mathbb{F}^2 \) and \( t: \mathbb{F} \times \mathbb{F}^3 \rightarrowtail \mathbb{F}^3 \) both be bilinear maps corresponding to scaling by \( \mathbb{F} \)
\( s \otimes t \) is a bilinear map from \( (\mathbb{F}^2 \otimes \mathbb{F}) \times (\mathbb{F} \otimes \mathbb{F}^3)\) to \((\mathbb{F}^2 \otimes \mathbb{F}^3) \)
\( s \otimes t \cong r \), for \(r: \mathbb{F}^2 \times \mathbb{F}^3 \rightarrowtail \mathbb{F}^6 \) the outer product tensor
Example
(s \otimes t)
\left(
\begin{bmatrix}u_1 \\ u_2 \end{bmatrix}
\otimes c_1,
c_2 \otimes
\begin{bmatrix}v_1 \\ v_2 \\ v_3 \end{bmatrix}
\right)
=
s
\left(
\begin{bmatrix} u_1 \\ u_2 \end{bmatrix},
c_2
\right)
\otimes
t
\left(
c_1,
\begin{bmatrix}v_1 \\ v_2 \\ v_3 \end{bmatrix}
\right)
= c_1c_2 \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \otimes \begin{bmatrix}v_1 & v_2 & v_3 \end{bmatrix} =
c_1c_2
\begin{bmatrix}
u_1v_1 & u_1v_2 & u_1v_3 \\
u_2v_1 & u_2v_2 & u_2v_3
\end{bmatrix}
Axis action equalities
Let \(t: U \times V \rightarrowtail W\). For \(\alpha \in \operatorname{End}(U), \beta \in \operatorname{End}(V), \gamma \in \operatorname{End}(W)\), axis actions are
$$\mathcal{L}(t) = \{(\alpha, \gamma) : t \bullet_2 \alpha = t \bullet_0 \gamma \}$$
$$\mathcal{M}(t) = \{(\alpha, \beta) : t \bullet_2 \alpha = t \bullet_1 \beta \}$$
$$\mathcal{R}(t) = \{(\beta, \gamma) : t \bullet_1 \beta = t \bullet_0 \gamma \}$$
$$\mathcal{C}(t) = \{(\alpha, \beta, \gamma) : t \bullet_2 \alpha = t \bullet_1 \beta = t \bullet_0 \gamma \}$$
$$\operatorname{Der}(t) = \{(\alpha, \beta, \gamma) : t \bullet_2 \alpha + t \bullet_1 \beta = t \bullet_0 \gamma \}$$
left nucleus
mid nucleus (adjoint)
right nucleus
centroid
derivation
t \bullet_2 \alpha: U \times V \rightarrowtail W \\ (u,v) \mapsto t(\alpha(u), v)
t \bullet_1 \beta: U \times V \rightarrowtail W \\ (u,v) \mapsto t(u, \beta(v))
t \bullet_0 \gamma: U \times V \rightarrowtail W \\ (u,v) \mapsto \gamma(t(u, v))
Results for nuclei
Theorem C (L.)
Let \(s: U_2 \times U_1 \rightarrowtail U_0\) and \(t: V_2 \times V_1 \rightarrowtail V_0 \) be fully non-degenerate bilinear maps. Then
$$\mathcal{L}(s \otimes t) = \mathcal{L}(s) \boxtimes \mathcal{L}(t)$$
$$\mathcal{M}(s \otimes t) = \mathcal{M}(s) \boxtimes \mathcal{M}(t)$$
$$\mathcal{R}(s \otimes t) = \mathcal{R}(s) \boxtimes \mathcal{R}(t)$$
Definition (\( \boxtimes \))
Let \(a = (\alpha_2, \alpha_1, \alpha_0) \), and \(b = (f_2, f_1, f_0) \), where \( \alpha_i \in \operatorname{End}(U_i) \), and \(f_i \in \operatorname{End}(V_i) \)
Corollary (Wilson Propositions 7.8 and 7.9, "Decomposing \(p\)-groups via Jordan Algebras")
Let \(d: U \times U \rightarrowtail C\) be a nondegenerate Hermitian \(C\)-form and \(b: V \times V \rightarrowtail W \) a \(k\)-bilinear map. Then \( \operatorname{Adj}(d \otimes b) = \operatorname{Adj}(d) \otimes \operatorname{Adj}(b) \).
Define \(a \boxtimes b \coloneqq (\alpha_2 \otimes f_2, \alpha_1 \otimes f_1, \alpha_0 \otimes f_0) \), where \(\alpha_i \otimes f_i \in \operatorname{End}(U_i \otimes V_i) \)
(s \otimes t) \bullet_2 x
=
\underbrace{
\left[\!\!\!\!
\begin{array}{c|c}
\phantom{0} & \phantom{0} \\
s\!\bullet_2\!\alpha_1 \!\! & \!\! s\!\bullet_2\!\alpha_2 \\
\phantom{0} & \phantom{0}
\end{array}
\!\!\!\!\right]
}_{\displaystyle A}
\underbrace{
\left[
\begin{array}{ccc}
\phantom{0} & t \bullet_2 f_1 & \phantom{0} \\
\hline
\phantom{0} & t \bullet_2 f_2 & \phantom{0}
\end{array}
\right]
}_{\displaystyle X}
=
\underbrace{
\left[\!\!\!\!
\begin{array}{c|c}
\phantom{0} & \phantom{0} \\
s\!\bullet_1\!\beta_1 \!\! & \!\! s\!\bullet_1\!\beta_2 \\
\phantom{0} & \phantom{0}
\end{array}
\!\!\!\!\right]
}_{\displaystyle B}
\underbrace{
\left[
\begin{array}{ccc}
\phantom{0} & t \bullet_1 g_1 & \phantom{0} \\
\hline
\phantom{0} & t \bullet_1 g_2 & \phantom{0}
\end{array}
\right]
}_{\displaystyle Y}
=
(s \otimes t) \bullet_1 y
Suppose \( (x,y) \in \mathcal{M}(s \otimes t) \), meaning \( (s \otimes t) \bullet_2 x = (s \otimes t) \bullet_1 y \)
Powered by isomorphism \(\operatorname{Bil}(U_2 \otimes V_2, U_1 \otimes V_1; U_0 \otimes V_0) \cong \operatorname{Bil}(U_2, U_1; U_0) \otimes \operatorname{Bil}(V_2, V_1; V_0) \)
Decompose \( x = \alpha_1 \otimes f_1 + \alpha_2 \otimes f_2 \) and \( y = \beta_1 \otimes g_1 + \beta_2 \otimes g_2 \)
\underbrace{
\left[\!\!\!\!
\begin{array}{c|c}
\phantom{0} & \phantom{0} \\
s\!\bullet_2\!\alpha_1 \!\! & \!\! s\!\bullet_2\!\alpha_2 \\
\phantom{0} & \phantom{0}
\end{array}
\!\!\!\!\right]
}_{\displaystyle A} M
=
\left[\!\!\!\!
\begin{array}{c|c}
\phantom{0} & \phantom{0} \\
s\!\bullet_2\!\tilde{\alpha_1} \!\! & \!\! s\!\bullet_2\!\tilde{\alpha_2} \\
\phantom{0} & \phantom{0}
\end{array}
\!\!\!\!\right]
=
\underbrace{
\left[\!\!\!\!
\begin{array}{c|c}
\phantom{0} & \phantom{0} \\
s\!\bullet_1\!\beta_1 \!\! & \!\! s\!\bullet_1\!\beta_2 \\
\phantom{0} & \phantom{0}
\end{array}
\!\!\!\!\right]
}_{\displaystyle B}
\( (\tilde{\alpha}_i, \beta_i) \in \mathcal{M}(s) \)
M^{-1}
\underbrace{
\left[
\begin{array}{ccc}
\phantom{0} & t \bullet_2 f_1 & \phantom{0} \\
\hline
\phantom{0} & t \bullet_2 f_2 & \phantom{0}
\end{array}
\right]
}_{\displaystyle X}
=
\left[
\begin{array}{ccc}
\phantom{0} & t \bullet_2 \tilde{f_1} & \phantom{0} \\
\hline
\rule{0pt}{1.1em} \phantom{0} & t \bullet_2 \tilde{f_2} & \phantom{0}
\end{array}
\right]
=
\underbrace{
\left[
\begin{array}{ccc}
\phantom{0} & t \bullet_1 g_1 & \phantom{0} \\
\hline
\phantom{0} & t \bullet_1 g_2 & \phantom{0}
\end{array}
\right]
}_{\displaystyle Y}
\( (\tilde{f}_i, g_i) \in \mathcal{M}(t) \)
\underbrace{
\begin{bmatrix}
3&1\\4&2
\end{bmatrix}
}_{\displaystyle M^{-1}}
\underbrace{
\left[
\begin{array}{cc}
1 & 2 & 0\\
\hline
0 & 1 & 1
\end{array}
\right]
}_{\displaystyle X}
=
\underbrace{
\begin{bmatrix}
3 & 2 & 1\\
\hline
4 & 0 & 2
\end{bmatrix}
}_{\displaystyle Y}
\(\exists M\) where \(AM = B\) and \(M^{-1}X = Y\)
\underbrace{
\left[
\begin{array}{c|c}
1 & 0 \\
2 & 1 \\
0 & 1
\end{array}
\right]
}_{\displaystyle A}
\underbrace{
\begin{bmatrix}
1&2\\3&4
\end{bmatrix}
}_{\displaystyle M}
=
\underbrace{
\left[
\begin{array}{c|c}
1 & 2\\
0 & 3\\
3 & 4
\end{array}
\right]
}_{\displaystyle B}
\(AX = BY \) minimal rank factorizations
A
=
\left[
\begin{array}{c|c}
1 & 0 \\
2 & 1 \\
0 & 1
\end{array}
\right]
X
=
\left[
\begin{array}{ccc}
1 & 2 & 0 \\
\hline
0 & 1 & 1
\end{array}
\right]
B
=
\left[
\begin{array}{c|c}
1 & 2 \\
0 & 3 \\
3 & 4
\end{array}
\right]
Y
=
\left[
\begin{array}{ccc}
3 & 2 & 1 \\
\hline
4 & 0 & 2
\end{array}
\right]
Upshot: global equality leads to local equalities
Field is \( \mathbb{F}_5 \)
\( \operatorname{col}(A)=\operatorname{col}(AX)=\operatorname{col}(BY)=\operatorname{col}(B) \)
If all nonzero, then up to scalars, either
$$a=b=c \text{ and } x+y=z$$
or
$$a+b=c \text{ and } x=y=z$$
(Grad student's dream)
\( \operatorname{Der}(s \otimes t) = \operatorname{Der}(s) \boxtimes \operatorname{Der}(t) \)
But alas, not the case.
Consider \(a \otimes x + b \otimes y = c \otimes z\), a triple in \( \operatorname{Der}(s \otimes t) \)
In \( \operatorname{Der}(s) \boxtimes \operatorname{Der}(t) \) means that both
- \( (a,b,c) \in \operatorname{Der}(s) \), so \( (a+b=c) \)
- \( (x,y,z) \in \operatorname{Der}(t)\), so \( (x+y=z) \)
Why?
Must have shared factor for sum of two pure tensors to be rank 1
Instead, term in either \( \mathcal{C}(s) \boxtimes \operatorname{Der}(t) \) or \( \operatorname{Der}(s) \boxtimes \mathcal{C}(t) \)
The nuclei case applies when one term is zero
Theorem D (L.)
Let \(s: U_2 \times U_1 \rightarrowtail U_0\) and \(t: V_2 \times V_1 \rightarrowtail V_0 \) be fully non-degenerate bilinear maps.
Then
\operatorname{Der}(s \otimes t) =
\operatorname{Der}(s) \boxtimes \mathcal{C}(t) +
\mathcal{C}(s) \boxtimes \operatorname{Der}(t) + \\
\qquad \qquad \qquad \qquad
\iota_{20}(\mathcal{L}(s) \boxtimes \mathcal{L}(t)) +
\iota_{10}(\mathcal{R}(s) \boxtimes \mathcal{R}(t)) +
\iota_{21}(\mathcal{M}(s) \boxtimes \mathcal{M}(t))
\\[0.5em]
\qquad \qquad \qquad \qquad
\scriptsize{\text{where } \iota_{20}(\alpha, \gamma) = (\alpha, 0, \gamma),
\iota_{10}(\beta,\gamma) = (0,\beta,\gamma),
\iota_{21}(\alpha, \beta) = (-\alpha, \beta, 0)}
Results for derivation
Corollary (Benkart-Osborn, Corollary 4.9, Derivations and Automorphisms of Nonassociative Matrix Algebras)
Let \(A\) be a unital algebra. Then \( \operatorname{Der}(\mathbb{M}_n(A)) = I_n \otimes \operatorname{Der}(A) + \operatorname{ad}(\mathbb{M}_n(\operatorname{Nuc}(A))) \)
Recall for \(u \in \operatorname{End}(A)\), the adjoint action \(\operatorname{ad}(u) \in \operatorname{Der}(A) \) is \( x \mapsto ux - xu \)
A Locally Independent Unified (LIU)-decomposition of \( (p,q,r) \) consists of a natural number \(n\) and vectors \(a_i \in A, x_i \in X, b_i \in B, y_i \in Y, c_i \in C, z_i \in Z\) such that
- (Decomposition Equality) \( p = \sum_{i=1}^{n} a_i \otimes x_i\), \(q = \sum_{i=1}^{n} b_i \otimes y_i \), and \(r = \sum_{i=1}^{n} c_i \otimes z_i \)
- (Local Equality) \(\forall i \in [n]\), \(a_i \otimes x_i + b_i \otimes y_i = c_i \otimes z_i \)
Key definition
p=\color{blue}{a_1\otimes x_1}+\color{green}{a_2\otimes x_2},
\qquad
\color{black}{q}=\color{blue}{b_1\otimes y_1}+\color{green}{b_2\otimes y_2},
\qquad
\color{black}{r}=\color{blue}{c_1\otimes z_1}+\color{green}{c_2\otimes z_2}.
Example
\(p+q=r\), with
\color{blue}{a_1\otimes x_1+b_1\otimes y_1=c_1\otimes z_1},
\qquad
\color{green}{a_2\otimes x_2+b_2\otimes y_2=c_2\otimes z_2}.
Local equalities:
p
=
\underbrace{
\left[\!\!\!\!
\begin{array}{c|c}
\phantom{0} & \phantom{0} \\
s\!\bullet_2\!\alpha_1 \!\! & \!\! s\!\bullet_2\!\alpha_2 \\
\phantom{0} & \phantom{0}
\end{array}
\!\!\!\!\right]
}_{\displaystyle A}
\underbrace{
\left[
\begin{array}{ccc}
\phantom{0} & t \bullet_2 f_1 & \phantom{0} \\
\hline
\phantom{0} & t \bullet_2 f_2 & \phantom{0}
\end{array}
\right]
}_{\displaystyle X}
Definition
Let \(p \in A \otimes X \leq U \otimes V\), \(q \in B \otimes Y \leq U \otimes V\), and \(r \in C \otimes Z \leq U \otimes V\) satisfy \(p+q=r\).
Lemma: Let \( (p,q,r) \) be a triple satisfying \(p+q=r\). Then \( (p,q,r) \) has a LIU-decomposition.
Idea: Decompose \(U = \bigoplus_{i=1}^{6} U^{(i)} \) and \(V = \bigoplus_{i=1}^{6} V^{(i)} \)
in aligned basis
\(p+q=r\) over \( \mathbb{F}_{997} \)
Fix to Grad Student's Dream
Theorem: There exists examples for \(p \in A \otimes X \leq U \otimes V\), \(q \in B \otimes Y \leq U \otimes V\), \(r \in C \otimes Z \leq U \otimes V\), and \(s \in D \otimes W \leq U \otimes V\) satisfying \(p+q+r\) without a LIU-decomposition.
Proof:
Work over \( \mathbb{F} = \mathbb{F}_5 \), and let \(U,V = \mathbb{F}^4 \), with standard bases
\( D = \text{span}\{e_1,e_2\} \leq U, W = \text{span}\{f_1,f_2\} \leq V \)
\( A = \text{span}\{e_1 + e_3, e_2 + e_4 \}, B =\text{span}\{e_1 - e_3, e_2 - e_4 \}, C = \text{span}\{e_1 + e_4, e_2 + 4e_3+ 4e_4 \} \)
\( X = \text{span}\{f_1 + f_3, f_2 + f_4 \}, Y =\text{span}\{f_1 - f_3, f_2 - f_4 \}, Z = \text{span}\{f_1 + 4f_4, f_2 + f_3+ 4f_4 \} \)
As notation, we shall denote \(A = \text{span}\{e_1+e_3 \eqqcolon a_1, e_2+e_4 \eqqcolon a_2\}, B = \text{span}\{b_1,b_2\}\), and so on
Then let
- \( p = a_1 \otimes x_1 + a_1 \otimes x_2 + 4 a_2 \otimes x_1 + 2 a_2 \otimes x_2 \)
- \( q = 3 b_1 \otimes y_1 + 2 b_1 \otimes y_2 + 3 b_2 \otimes y_1\)
- \( r = 2 c_1 \otimes z_1 + 2 c_1 \otimes z_2 + 3c_2 \otimes z_1 + 4 c_2 \otimes z_2 \)
- \( s = e_1 \otimes f_1 + e_2 \otimes f_2 \)
This quadruple satisfies \(p+q+r = s\), with \(p \in A \otimes X, q \in B \otimes Y, r \in C \otimes Z, s \in D \otimes W \), and no local equality exists because any nonzero element of \((A \otimes X + B \otimes Y + C \otimes Z) \cap (D \otimes W)\) is rank 2. (Magma calculation)
This example was created so elements in the intersection has to satisfy
\begin{bmatrix}
P+Q+R = S & P-Q+RJ = 0 \\
P-Q+JR = 0 & P+Q+JRJ = 0
\end{bmatrix}
And choosing \(J\) such that \(S \in \mathbb{F}[J] \) is invertible
SKIP DUE TO TIME
If \((x,y) \in \mathcal{M}(s \otimes t) \), the equation \( (s \otimes t)\bullet_2 x = (s \otimes t) \bullet_1 y \) is matrix equality by \( (*) \)
For \(AX = (s \otimes t) \bullet_2 x = (s \otimes t) \bullet_1 y = BY\) minimal rank factorizations, each column of \(A\) is in \(\operatorname{CS}(B) \), and each row of \(X\) is in \(\operatorname{RS}(Y)\)
Key idea for nuclei proof
Extras that I cut
Corollary (Brešar, Theorem 3.1, Derivations of Tensor Products of Nonassociative Algebras)
Let \(R\) and \(S\) be non-associative algebras.
Then every derivation of \(R \otimes S\) can be written as \(d = \operatorname{ad}(u) + \sum_{j=1}^{p} \lambda_{z_j} \otimes f_j + \sum_{i=1}^{q} g_i \otimes \lambda_{w_i} \), where \(u \in \operatorname{Nuc}(R) \otimes \operatorname{Nuc}(S) \), \(z_j \in Z(R)\), \(f_j \in \operatorname{Der}(S) \), and \(g_i \in \operatorname{Der}(R) \)
Corollary - \( \mathcal{C}(s \otimes t) = \mathcal{C}(s) \boxtimes \mathcal{C}(t) \)
In summary
- Tensors are multilinear maps whose algebraic invariants discern structure
- Theorem A, Theorem B - faster algorithm for adjoint and derivation
- Theorem C, Theorem D - decomposition theorems on the nuclei and derivation
\( I \)
\( J \)
\( M_R \)
\( M_S \)
- Algorithms for adjoint and derivation algebra of higher valence tensors (e.g trilinear maps)
- Weaken regular family and black-box restriction requirements using randomized computation model
- Decomposing the product of bilinear maps
Future directions
Questions?
- Tensors are multilinear maps whose algebraic invariants discern structure
- Theorem A, Theorem B - faster algorithm for adjoint and derivation
- Theorem C, Theorem D - decomposition theorems on the nuclei and derivation
Computing Algebraic Invariants of Tensors and Their Products
By Chris Liu
