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reference updated to peer-reviewed version
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Tim Seppelt
Tim Seppelt
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A sum-of-entries version of Specht's theorem was proven in Theorem 4.3 of Grohe et al. (2021): Homomorphism Tensors and Linear EquationsTheorem 20 of Grohe et al. (2022): Homomorphism Tensors and Linear Equations. The condition you're describing is also sufficient. More precisely, for matrices $A$ and $B$ it holds that $\sigma(w(A,A^*)) = \sigma(w(B,B^*))$ if and only if there exists a pseudo-stochastic matrix $X$ such that $XA = BX$ and $XA^* = B^*X$. A matrix is called pseudo-stochastic if $X \boldsymbol{1} = \boldsymbol{1} = X^T \boldsymbol{1}$ for $\boldsymbol{1}$ the all-ones vector.

A sum-of-entries version of Specht's theorem was proven in Theorem 4.3 of Grohe et al. (2021): Homomorphism Tensors and Linear Equations. The condition you're describing is also sufficient. More precisely, for matrices $A$ and $B$ it holds that $\sigma(w(A,A^*)) = \sigma(w(B,B^*))$ if and only if there exists a pseudo-stochastic matrix $X$ such that $XA = BX$ and $XA^* = B^*X$. A matrix is called pseudo-stochastic if $X \boldsymbol{1} = \boldsymbol{1} = X^T \boldsymbol{1}$ for $\boldsymbol{1}$ the all-ones vector.

A sum-of-entries version of Specht's theorem was proven in Theorem 20 of Grohe et al. (2022): Homomorphism Tensors and Linear Equations. The condition you're describing is also sufficient. More precisely, for matrices $A$ and $B$ it holds that $\sigma(w(A,A^*)) = \sigma(w(B,B^*))$ if and only if there exists a pseudo-stochastic matrix $X$ such that $XA = BX$ and $XA^* = B^*X$. A matrix is called pseudo-stochastic if $X \boldsymbol{1} = \boldsymbol{1} = X^T \boldsymbol{1}$ for $\boldsymbol{1}$ the all-ones vector.

Source Link
Tim Seppelt
Tim Seppelt
  • 58
  • 5

A sum-of-entries version of Specht's theorem was proven in Theorem 4.3 of Grohe et al. (2021): Homomorphism Tensors and Linear Equations. The condition you're describing is also sufficient. More precisely, for matrices $A$ and $B$ it holds that $\sigma(w(A,A^*)) = \sigma(w(B,B^*))$ if and only if there exists a pseudo-stochastic matrix $X$ such that $XA = BX$ and $XA^* = B^*X$. A matrix is called pseudo-stochastic if $X \boldsymbol{1} = \boldsymbol{1} = X^T \boldsymbol{1}$ for $\boldsymbol{1}$ the all-ones vector.