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linear algebra – Generalizing the Pfaffian: households of matrices whose determinants are consummate powers of polynomials within the entries

A very good class of examples of that is given by Clifford algebras: If $V$ is a actual vector house with endowed with a quadratic figure $q:Vtomathbb{R}$, the algebra $Cl(q)$ is the algebra generated by the weather of $V$ matter to the multiplication rule $x^2 = -q(x)$. If $M$ is a $Cl(q)$-module, say $Msimeqmathbb{R}^m$, then we now have an inclusion $Vhookrightarrowmathrm{End}(M)$ and the attribute polynomial of $xin Vsubseteqmathrm{End}(M)$ is well seen to breathe $(t^2+q(x))^{m/2}$, so we now have

$$

det(x) = q(x)^{m/2}

$$

for all $xin V$.

For instance, if $V$ is $mathbb{R}^8$ with its benchmark Euclidean quadratic figure $q$, then $Cl(q)$ is isomorphic to $mathrm{End}_{mathbb{R}}(mathbb{R}^{16})$, so we will take $M=mathbb{R}^{16}$ (and each $Cl(q)$-module is $mathbb{R}^{16k}$ for some integer $okay$). Thus, on this illustration, we now have $det(x) = p(x)^8$ the place $p(x) = |x|^2$ for all $xin V$.

In common, when $Vsimeqmathbb{R}^n$ and $q_n:Vtomathbb{R}$ is nondegenerate, the dimension of a minimal nontrivial $Cl(q_n)$-module grows (roughly) exponentially with $n$, so the minimal $m$ grows exponentially with $n$. This exhibits that there are nontrivial ‘irreducible’ examples with $det(x) = p(x)^okay$ for $okay$ arbitrarily sizable and there’s no certain on the workable dimension $n$ of the subspace $Vsubsetmathrm{End}(M)$.

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