# Generalizing the Pfaffian: families of matrices whose determinants are perfect powers of polynomials in the entries

Let $$n$$ be a positive integer, and let $$M = (m_{ij})$$ be a skew $$2n \times 2n$$ matrix. That is, we have $$m_{ij} = -m_{ji}$$ for $$1 \leq i, j \leq 2n$$. Then it is well-known that

$$\det M = p(M)^2,$$

where $$p$$ is a polynomial in the entries $$m_{ij}$$. The polynomial $$p(M)$$ is called the Pfaffian of $$M$$.

Is there a generalization of this? That is, is there a natural family of $$kn \times kn$$ matrices whose determinants are perfect $$k$$-th powers of polynomials in the entries?

• Singular matrices. :-) – LSpice Jan 9 at 0:57
• – Libli Jan 9 at 6:31
• @Libli The $n = 4m$ case seems closely related to what I need... if you could expand on this and give a reference I will accept that as the answer – Stanley Yao Xiao Jan 9 at 7:51
• Block matrices with an $n \times n$ block repeated $k$ times :-) – Zach Teitler Jan 9 at 21:01

A good class of examples of this is given by Clifford algebras: If $$V$$ is a real vector space with endowed with a quadratic form $$q:V\to\mathbb{R}$$, the algebra $$Cl(q)$$ is the algebra generated by the elements of $$V$$ subject to the multiplication rule $$x^2 = -q(x)$$. If $$M$$ is a $$Cl(q)$$-module, say $$M\simeq\mathbb{R}^m$$, then we have an inclusion $$V\hookrightarrow\mathrm{End}(M)$$ and the characteristic polynomial of $$x\in V\subseteq\mathrm{End}(M)$$ is easily seen to be $$(t^2+q(x))^{m/2}$$, so we have $$\det(x) = q(x)^{m/2}$$ for all $$x\in V$$.
For example, if $$V$$ is $$\mathbb{R}^8$$ with its standard Euclidean quadratic form $$q$$, then $$Cl(q)$$ is isomorphic to $$\mathrm{End}_{\mathbb{R}}(\mathbb{R}^{16})$$, so we can take $$M=\mathbb{R}^{16}$$ (and every $$Cl(q)$$-module is $$\mathbb{R}^{16k}$$ for some integer $$k$$). Thus, in this case, we have $$\det(x) = p(x)^8$$ where $$p(x) = |x|^2$$ for all $$x\in V$$.
In general, when $$V\simeq\mathbb{R}^n$$ and $$q_n:V\to\mathbb{R}$$ 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 shows that there are nontrivial 'irreducible' examples with $$\det(x) = p(x)^k$$ for $$k$$ arbitrarily large and that there is no bound on the possible dimension $$n$$ of the subspace $$V\subset\mathrm{End}(M)$$.
Remark: Given a linear subspace $$V\subset\mathrm{End}(\mathbb{R}^{m})$$ such that there exists a polynomial $$p:V\to\mathbb{R}$$ and an integer $$k = m/\deg(p)>1$$ such that $$\det(x) = p(x)^k$$, we say that the pair $$(V,\mathbb{R}^m)$$ is irreducible if there is no nontrivial subspace $$M\subset\mathbb{R}^m$$ such that $$x(M)\subset M$$ for all $$x\in V$$ and $$\det(x_{|M}) = p(x)^j$$ for all $$x\in V$$, where, necessarily, $$j = (\dim M)/\deg(p)$$.
The interesting problem for linear subspaces $$V\subset\mathrm{End}(\mathbb{R}^m)$$ on which the $$\det$$-function is a higher power of a polynomial on $$V$$ is to classify the irreducible ones of maximal dimension for a given $$m$$.