Mixed ladder determinantal varieties from two-sided ladders

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…

Computation of the $a$-invariant of ladder determinantal rings

We solve the problem of effectively computing the $a$-invariant of ladder determinantal rings. In the case of a one-sided ladder, we provide a compact formula, while, for a large family of two-sided ladders, we provide an algorithmic solution.

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。

Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of cert…

Gröbner bases and the Cohen-Macaulay property of Li's double determinantal varieties

We consider double determinantal varieties, a special case of Nakajima quiver varieties. Li conjectured that double determinantal varieties are normal, irreducible, Cohen-Macaulay varieties whose defining ideals have a Gröbner basis given by their natural generators. We use liaison theory to prove this conjecture in a manner that generalizes results for mixed ladder determinantal varieties. We also give a formula for the dimension of a double determinantal variety.

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.

行列式模的爆破代数

Blowup algebras of determinantal modules

本文研究由阶梯矩阵极大子式或单位区间行列式理想生成的模的爆破代数，确定其多重Rees代数及特殊纤维环的Gröbner基，并证明所得代数为Koszul Cohen–Macaulay正规整环，且在特征零时具有有理奇点、正特征时为F-有理。

关于分块行列式理想的约化 Gröbner 基

On the Reduced Gröbner Bases of Blockwise Determinantal Ideals

本文针对分块行列式理想，建立了验证其关于（反）对角项序的 Gröbner 基为极小或约化的准则，并特别给出了舒伯特行列式理想在非 Vexillary 情形下由 elusive 子式直接计算约化 Gröbner 基的显式公式，同时证明了相关 W-特征集和特征对的正常性与强性。

Rigid ideals by deforming quadratic letterplace ideals

We compute the deformation space of quadratic letterplace ideals $L(2,P)$ of finite posets $P$ when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal $J(2,P)$ defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases $J(2,P)$ is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.

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。

Gröbner基、对称矩阵与C型Kazhdan-Lusztig簇

Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties

研究一般对称矩阵子式生成的多项式理想，作为C型Kazhdan-Lusztig簇的概型定义理想，给出了其Gröbner基、初始理想素分解及基于pipe dream的Hilbert级数公式。

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 …