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Recall that a homogeneous polynomial $P\in{\mathbb R}[X_1,\ldots,X_d]$ of degree $n$ is hyperbolic in the direction of a vector $V\ne0$ if for every vector $W$, the univariate polynomial $t\mapsto P(W+tV)$ has $n$ real roots. Notice that $P(V)$ must be non-zero, and we may assume w.l.o.g. that $P(V)>0$. The notion of hyperbolic polynomials is due to L. Garding, because of hyperbolic PDEs. He proved, among other things, that the connected component of $V$ in $\{P(W)>0\}$ is a convex cone (the cone of future in terms of evolutionary PDEs), and that $P$ is hyperbolic in every direction $V'$ of this cone. This is reminiscent to special relativity.

Examples of hyperbolic polynomials include

  • $P=X_1\cdots X_d$, where $n=d$ and $V$ can be taken as $(1,\ldots,1)$
  • $P(S)=\det S$, where $d=\frac{n(n+1)}2$ and ${\mathbb R}^d$ identifies with the space of symmetric matrices with real entries. Then $V=I_n$ and one uses the fact that eigenvalues of symmetric matrices are real. One may replace real symmetric by complex hermitian matrices, with $d=n^2$ instead.

Besides, let me recall a result of I. Schur about (generalized) permanents: let $H$ be a subgroup of ${\frak S}_n$, and $\chi$ be a complex character over $H$. Then, for every $S\in SPD_n$, one has $$\chi(e)\det S\le d_\chi(S):=\sum_{g\in H}\chi(g)\prod_{i=1}^ns_{ig(i)}.$$ In particular, the connected component of $I_n$ in $d_\chi>0$ contains the cone $SPD_n$.
This suggest that some of the $d_\chi$'s could be hyperbolic in the direction of $I_n$. This is true at least in the following cases:

  • $H=\frak S_n$ and $\chi=\epsilon$ the signature, where $d_\chi=\det$,
  • $H=(e)$, where $d_\chi(S)=\prod_i s_{ii}$ (remark that the corresponding Schur's inequality is known has Hadamard inequality),

    In between, when $H$ is the natural inclusion of $\frak S_m\times\cdots\times\frak S_q$ with $M+\cdots+q\le n$, and $\chi$ is the product of the signatures, then $d_\chi$ is the product of hyperbolic polynomials of the types above, hence is itself hyperbolic.

However, the choice $H={\frak S}_n$ and $\chi={\bf 1}$ yields the permanent, which is not hyperbolic in the direction of $I_n$ if $n\ge2$.

My question is whether there are other hyperbolic polynomials among the $d_\chi$'s. If so, how do they classify ? Does hyperbolicity correspond to a classical property of $(H,\chi)$ ?

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