分次默比乌斯代数上的径向导数
The radial derivative on the graded Möbius algebra
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中文总结 AI 辅助
该研究针对简单拟阵的分次默比乌斯代数构造了径向导数算子$D_\beta$，证明了其希尔伯特级数$H_\beta$不是赋值拟阵不变量，验证了相关对数凹性猜想。
中文摘要 AI 辅助
设$M$是一个简单拟阵，$B(M)$是其平集格的分次默比乌斯代数。平集的有序基权重定义了一个内积，在基多项式实现下，原子乘法的伴随算子成为普通坐标导数。由此，我们构造了一个典范的全局下推算子$D_\beta$，它在典范“径向”截断多项式代数副本上表现为普通微分。让$D_\beta$和坐标导数共同作用，会产生一个分次循环模，其希尔伯特级数为$H_{\beta,M}(q)=\sum_{k=0}^r h_k^\beta(M)q^k$。我们给出了具有相同Derksen $\mathcal{G}$不变量和相同经典互反希尔伯特级数，但$H_\beta$不同的拟阵例子，因此$H_\beta$不能是赋值拟阵不变量在简单拟阵上的限制。我们猜想$H_\beta$在微分次数上是对数凹且顶部重的。对于包含Larson对Whitney对数凹性的反例的广义theta族，我们计算了前四个系数并证明了关键的对数凹性不等式。对8个元素上的所有950个简单拟阵的精确计算验证了这两个猜想。
英文摘要
Let $M$ be a simple matroid and let $B(M)$ be the graded Möbius algebra of its lattice of flats. The ordered-basis weights of flats define an inner product for which the adjoints of atom multiplication become ordinary coordinate derivatives under the basis-polynomial realization. From this, we construct a canonical global lowering operator $D_β$ which acts as ordinary differentiation on a canonical ``radial'' copy of a truncated polynomial algebra. Allowing both $D_β$ and the coordinate derivatives to act produces a graded cyclic module with Hilbert series \[ H_{β,M}(q)=\sum_{k=0}^r h_k^β(M)q^k. \] We give examples of matroids with the same Derksen $\mathcal G$-invariant and the same classical apolar Hilbert series but different $H_β$. Hence $H_β$ cannot be the restriction to simple matroids of a valuative matroid invariant. We conjecture that $H_β$ is log-concave and top-heavy in differential degree. For the generalized theta family containing Larson's counterexample to Whitney log-concavity, we compute the first four coefficients and prove the critical log-concavity inequality. Exact computation verifies both conjectures for all $950$ simple matroids on eight elements.
