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

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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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