Coxeter Groups and Unimodality — Conjecture 15

Definitions and mathematical statement

The supplied excerpt does not identify the matrix whose submatrices occur in the conjecture. Consequently, only the following statement schema can be recorded; it is not a complete mathematical formulation.

Write $A=(a_{p,q})$ for the unspecified matrix. For $i,j,r\in\mathbb{N}$, define, whenever the indicated entries exist, \[ D_{j,r}(i)=\det\left(a_{i+s,j+t}\right)_{0\le s,t\le r}. \] Thus $D_{j,r}(i)$ is the determinant of the $(r+1)\times(r+1)$ submatrix with consecutive rows $i,i+1,\ldots,i+r$ and consecutive columns $j,j+1,\ldots,j+r$.

The assertion attributed to Brenti, Conjecture 15 (2024), is that, for every fixed $j,r\in\mathbb{N}$, the function $i\mapsto D_{j,r}(i)$ is monotonically increasing on $\mathbb{N}$. If increasing means nondecreasing, the asserted inequalities are \[ D_{j,r}(i+1)\ge D_{j,r}(i) \qquad\text{for all }i,j,r\in\mathbb{N}. \] If strict increase is intended, $\ge$ must instead be replaced by $>$.

The matrix, its entries and their domain, any additional parameters or hypotheses, and the ordering in which determinants are compared are absent from the supplied record. The convention for whether $0\in\mathbb{N}$ is also unspecified; it affects both the allowed starting indices and whether $r=0$ is included. These omissions must be resolved before the conjecture has a precise mathematical target.

Short English statement

For every fixed column starting index $j$ and block size $r+1$, the conjecture asserts that the determinant of a consecutive square block increases as its row starting index $i$ increases. The supplied excerpt omits the underlying matrix and does not clarify whether increase is strict, so the statement remains incomplete.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800377$, exact source problem code catalogue entry, title "Coxeter Groups and Unimodality --- Conjecture 15." The record attributes the conjecture to Brenti (2024) and labels it open; that status is supplied curation, not an independently verified mathematical assessment.
• Brenti, Some Open Problems on Coxeter Groups and Unimodality (2024), Conjecture 15. Supplied source URL: https://arxiv.org/abs/2410.09897. The supplied background records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; no independent source visit or verification is claimed.