# Hyperbolic polynomials and group representations

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 $(H,\chi)$ ?

[EDIT]: Recently, I ran into this question again, so decided to undelete this 3 year old answer of mine.

Although not an answer to the exact question, I thought it worth mentioning an important problem where characterizing certain situations where the permanent leads to a hyperbolic polynomial was an open problem for many years. Details follow below.

(Defn.) Let $A$ be a square real matrix. We say that an $n \times n$ real matrix $A$ is column monotone if it has weakly decreasing entries in each column (when reading downwards). That is, $a_{ij} \ge a_{i+1,j}$ for all $1 \le i \le n-1$ and $1\le j \le n$.

Based on this definition, (MCPC) The Monotone Column Permanent Conjecture of Haglund, Ono, Wagner ("Theorems and conjectures involving rook polynomials with only real zeros", Kluwer, 1999), is as follows.

(Erstwhile conjecture). If $A$ is an $n \times n$ column monotone matrix, then $\text{per}(A+zJ)$ is a polynomial in $z$ that has only real roots, where the direction $J$ is an $n \times n$ matrix of all ones.

This conjecture was proved in its full generality in "Proof of the Montone Column Permanent Conjecture" by Brändén, Haglund, Visontai, and Wagner (2010).

My summary above is taken from the abovecited paper. If I find anything related to immanants, I'll update my answer.