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Apoorva Khare and I have updated our paper “On the sign patterns of entrywise positivity preservers in fixed dimension“, announced at this post from last month. The quantitative results are now sharpened using a new monotonicity property of ratios ${s_{\lambda}(u)/s_{\mu}(u)}$ of Schur polynomials, namely that such ratios are monotone non-decreasing in each coordinate of ${u}$ if ${u}$ is in the positive orthant, and the partition ${\lambda}$ is larger than that of ${\mu}$. (This monotonicity was also independently observed by Rachid Ait-Haddou, using the theory of blossoms.) In the revised version of the paper we give two proofs of this monotonicity. The first relies on a deep positivity result of Lam, Postnikov, and Pylyavskyy, which uses a representation-theoretic positivity result of Haiman to show that the polynomial combination

$\displaystyle s_{(\lambda \wedge \nu) / (\mu \wedge \rho)} s_{(\lambda \vee \nu) / (\mu \vee \rho)} - s_{\lambda/\mu} s_{\nu/\rho} \ \ \ \ \ (1)$

of skew-Schur polynomials is Schur-positive for any partitions ${\lambda,\mu,\nu,\rho}$ (using the convention that the skew-Schur polynomial ${s_{\lambda/\mu}}$ vanishes if ${\mu}$ is not contained in ${\lambda}$, and where ${\lambda \wedge \nu}$ and ${\lambda \vee \nu}$ denotes the pointwise min and max of ${\lambda}$ and ${\nu}$ respectively). It is fairly easy to derive the monotonicity of ${s_\lambda(u)/s_\mu(u)}$ from this, by using the expansion

$\displaystyle s_\lambda(u_1,\dots, u_n) = \sum_k u_1^k s_{\lambda/(k)}(u_2,\dots,u_n)$

of Schur polynomials into skew-Schur polynomials (as was done in this previous post).

The second proof of monotonicity avoids representation theory by a more elementary argument establishing the weaker claim that the above expression (1) is non-negative on the positive orthant. In fact we prove a more general determinantal log-supermodularity claim which may be of independent interest:

Theorem 1 Let ${A}$ be any ${n \times n}$ totally positive matrix (thus, every minor has a non-negative determinant). Then for any ${k}$-tuples ${I_1,I_2,J_1,J_2}$ of increasing elements of ${\{1,\dots,n\}}$, one has

$\displaystyle \det( A_{I_1 \wedge I_2, J_1 \wedge J_2} ) \det( A_{I_1 \vee I_2, J_1 \vee J_2} ) - \det(A_{I_1,J_1}) \det(A_{I_2,J_2}) \geq 0$

where ${A_{I,J}}$ denotes the ${k \times k}$ minor formed from the rows in ${I}$ and columns in ${J}$.

For instance, if ${A}$ is the matrix

$\displaystyle A = \begin{pmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{pmatrix}$

for some real numbers ${a,\dots,p}$, one has

$\displaystyle a h - de\geq 0$

(corresponding to the case ${k=1}$, ${I_1 = (1), I_2 = (2), J_1 = (4), J_2 = (1)}$), or

$\displaystyle \det \begin{pmatrix} a & c \\ i & k \end{pmatrix} \det \begin{pmatrix} f & h \\ n & p \end{pmatrix} - \det \begin{pmatrix} e & h \\ i & l \end{pmatrix} \det \begin{pmatrix} b & c \\ n & o \end{pmatrix} \geq 0$

(corresponding to the case ${k=2}$, ${I_1 = (2,3)}$, ${I_2 = (1,4)}$, ${J_1 = (1,4)}$, ${J_2 = (2,3)}$). It turns out that this claim can be proven relatively easy by an induction argument, relying on the Dodgson and Karlin identities from this previous post; the difficulties are largely notational in nature. Combining this result with the Jacobi-Trudi identity for skew-Schur polynomials (discussed in this previous post) gives the non-negativity of (1); it can also be used to directly establish the monotonicity of ratios ${s_\lambda(u)/s_\mu(u)}$ by applying the theorem to a generalised Vandermonde matrix.

(Log-supermodularity also arises as the natural hypothesis for the FKG inequality, though I do not know of any interesting application of the FKG inequality in this current setting.)