A determinantal formula for the Hilbert series of one-sided ladder determinantal rings

On semidualizing modules of ladder determinantal rings

We identify all semidualizing modules over certain classes of ladder determinantal rings over a field ${\mathsf k}$. Specifically, given a ladder of variables $Y$, we show that the ring ${\mathsf k}[Y]/I_t(Y)$ has only trivial semidualizing modules up to isomorphism in the following cases: (1) $Y$ is a one-sided ladder, and (2) $Y$ is a two-sided ladder with $t=2$ and no coincidental inside corners.

阶梯行列式模的Gorenstein特纤维环

Gorenstein Special Fiber Rings of Ladder Determinantal Modules

本文研究了阶梯行列式模的特纤维环为Gorenstein的条件，通过分析底层矩阵的数据来确定其性质。

Semidualizing modules of $2 \times 2$ ladder determinantal rings

We continue our study of ladder determinantal rings over a field $\mathsf k$ from the perspective of semidualizing modules. In particular, given a ladder of variables $Y$, we show that the associated ladder determinantal ring $\mathsf k[Y]/I_2(Y)$ admits exactly $2^n$ non-isomorphic semidualizing modules where $n$ is determined from the combinatorics of the ladder $Y$: the number $n$ is essentially the number of non-Gorenstein factors in a certain decomposition of $Y$. From this, for each $n$, we show explicitly how to find ladders $Y$ such that $\mathsf k[Y]/I_2(Y)$ admits exactly $2^n$ non-i…

梯型行列式模的特殊纤维环的代数不变量

Algebraic invariants of special fiber rings of ladder determinantal modules

该研究给出梯型行列式模的特殊纤维环的关键代数不变量显式公式，通过Sagbi退化、Gröbner退化简化计算，其重数公式推广了格拉斯曼流形次数的经典公式。

Counting of paths and the multiplicity of determinantal rings

In this paper, we derive several formulas of counting families of non-intersecting paths for two-sided ladder-shaped regions. As an application, we give a new proof to a combinatorial interpretation of Fibonacci numbers obtained by G. Andrews in 1974.

Rees algebras of sparse determinantal ideals

We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a $2\times n$ sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic.

Almost Gorenstein Hibi rings

In this paper, we state criteria of a Hibi ring to be level, non-Gorenstein and almost Gorenstein and to be non-level and almost Gorenstein in terms of the structure of the partially ordered set defining the Hibi ring. We also state a criterion of a ladder determinantal ring defined by 2-minors to be non-Gorenstein and almost Gorenstein in terms of the shape of the ladder.

Veronese子代数的典范迹的双侧界

Two-sided bounds for canonical traces of Veronese subalgebras

本文推广Goto--Watanabe的Veronese公式至更广的模有限扩张，得到典范迹的双侧界，并刻画了近似Gorenstein性质的条件，应用于Segre积等环类。

Properties of the Toric Rings of a Chordal Bipartite Family of Graphs

This work concerns the study of properties of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain ladder-like structures). We determine a linear system of parameters for each ring and explicitly determine the Hilbert series for the resulting Artinian reduction. As corollaries, we obtain the multiplicity and regularity of the original rings. This work extends results easily derived from lattice theory for a subfamily coming from a two-sided ladder to a family …