Barvinok introduced the symmetrized determinant ($\sdet$) as a \emph{non-commutative} analogue of the determinant. Intuitively, given a square matrix over an associative algebra, we can obtain the symmetrized determinant by averaging over all possible multiplication orders in the Leibniz formula for the determinant. He used the symmetrized determinant to design algorithms estimating the permanent of a matrix. To this end, he showed that there is a $O(n^{r+3})$ algorithm computing $\sdet$, where $r$ is the dimension of the algebra, and is therefore polynomial-time computable for fixed $r$. In this work, we study the algebraic properties and complexity of $\sdet$. While most of the properties of the ordinary determinant don't generalize to $\sdet$ defined on non-commutative algebras, we show that the principal minor expansion of the $\sdet$ is analogous to the ordinary determinant. Second, we prove that there exists a polynomial-sized algebra such that computing the symmetrized determinant is $\sharpP$-hard. Third, we show that the associated polynomial family is $\VNP$-complete over a suitable polynomial-dimensional algebra in the non-commutative setting. Further, when seen as a family of polynomials over the matrix algebra, it is also $\VNP$-complete in the commutative setting. This places the symmetrized determinant among the natural complete families arising from algebraic computation.

翻译：Barvinok 引入对称化行列式（$\sdet$）作为行列式的非交换模拟。直观上，给定一个结合代数上的方阵，通过对莱布尼茨行列式公式中所有可能的乘法顺序进行平均，即可得到对称化行列式。他利用对称化行列式设计了估算矩阵积和式的算法。为此，他证明了存在一个 $O(n^{r+3})$ 的算法用于计算 $\sdet$，其中 $r$ 是代数的维数，因此对于固定的 $r$，该算法是多项式时间可计算的。在本工作中，我们研究 $\sdet$ 的代数性质与计算复杂性。尽管普通行列式的大多数性质无法推广到定义在非交换代数上的 $\sdet$，但我们证明 $\sdet$ 的主子式展开与普通行列式类似。其次，我们证明存在一个多项式大小的代数，使得计算对称化行列式是 $\sharpP$-困难的。第三，我们证明在非交换设定下，相关的多项式族在适当的多项式维数代数上是 $\VNP$-完全的。进一步地，当将其视为矩阵代数上的多项式族时，在交换设定下它也是 $\VNP$-完全的。这使得对称化行列式成为源于代数计算的自然完全族之一。