Hsin-Ju Wang
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.
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This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ a $m \times n$-matrix of variables, we utilize Gröbner bases to give a closed form the length $λ( k[X] / (I_2(X) + \mathfrak{m}^{ \lceil sq \rceil} + \mathfrak{m}^{[q]} ))$ where $s \in \mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $\mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {\it polynomial} function of $q$ for all $s$.
Herzog, Huneke, and Srinivasan have conjectured that for any homogeneous $k$-algebra, the multiplicity is bounded above by a function of the maximal degrees of the syzygies and below by a function of the minimal degrees of the syzygies. The goal of this paper is to establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field $k$ for $k$-algebras $k[x_1, ..., x_n]/I$ being $I$ a determinantal ideal of arbitrary codimension.
In this paper we investigate the question of when the determinantal ring $R$ over a field $k$ is an almost Gorenstein local/graded ring in the sense of Goto, Takahashi, and the author. As a consequence of the main result, we see that if $R$ is a non-Gorenstein almost Gorenstein local/graded ring, then the ring $R$ has a minimal multiplicity.
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 finite $O$-sequences of a given multiplicity
研究给定乘数d的有限O序列数量O_d，证明其序列是次斐波那契序列，并推导出计算O_d的元素方法及迭代公式。
Invariants of toric double determinantal rings
本文研究了由大小为2的子式生成的双重行列式理想$I_{mn}^r$，证明其等同于特定有限分配格的Hibi环，计算了环$R/I_{mn}^r$的多个代数不变量，刻画了其Gorenstein性质，并对初始理想的Stanley-Reisner复形面给出了组合描述。
Cohomology of flag supervarieties and resolutions of determinantal ideals. II
本文计算了复periplectic Grassmannian结构层的凝聚上同调，证明其可分解为辛或正交Grassmannian上同调环与periplectic Lie超群半单表示的张量积，并建立了与行列式理想syzygy的联系。
Let $S=\mathbb{C}[x_{ij}]$ be a polynomial ring of $m\times n$ generic variables (resp. a polynomial ring of $(2n+1) \times (2n+1)$ skew-symmetric variables) over $\mathbb{C}$ and let $I$ (resp. Pf) be the determinantal ideal of maximal minors (resp. sub-maximal pfaffians) of $S$. Using the representation theoretic techniques introduced in the work of Raicu et al, we study the asymptotic behavior of the length of the local cohomology module of determinantal and pfaffian thickenings for suitable choices of cohomological degrees. This asymptotic behavior is also defined as a notion of multiplict…
We give three determinantal expressions for the Hilbert series as well as the Hilbert function of a Pfaffian ring, and a closed form product formula for its multiplicity. An appendix outlining some basic facts about degeneracy loci and applications to multiplicity formulae for Pfaffian rings is also included.
Defining ideals of some numerical semigroup rings with arithmetic pseudo-Frobenius numbers
本文研究了数值半群环的定义理想，证明在特定条件下，当伪Frobenius数构成等差数列时，该理想为行列式型，并部分解决了相关猜想。
We answer some questions related to multiplicity formulas by Rosenthal and Zelevinsky and by Lakshmibai and Weyman for points on Schubert varieties in Grassmannians. In particular, we give combinatorial interpretations in terms of nonintersecting lattice paths of these formulas, which makes the equality of the two formulas immediately obvious. Furthermore we provide an alternative determinantal formula for these multiplicities, and we show that they count semistandard tableaux of unusual shapes.
The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay complete local ring $R$ with a parameter ideal $Q$, the Auslander-Reiten conjecture holds for $R$ if and only if it holds for the residue ring $R/Q$. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring $R/Q^\ell$ in connection with that for $R$, and prove the equivalence of them for the case where $R$ is Gorenstein and $\ell\le \dim R$. In the latter part, we generalize the result of the minimal multiplicity by J. Sally. Due to these two of our…
Kazhdan-Lusztig ideals, a family of generalized determinantal ideals investigated in [Woo-Yong '08], provide an explicit choice of coordinates and equations encoding a neighbourhood of a torus-fixed point of a Schubert variety on a type A flag variety. Our main result is a Grobner basis for these ideals. This provides a single geometric setting to transparently explain the naturality of pipe dreams on the Rothe diagram of a permutation, and their appearance in: * combinatorial formulas [Fomin-Kirillov '94] for Schubert and Grothendieck polynomials of [Lascoux-Schutzenberger '82]; * the equivar…
Frobenius multiplicities and F-signatures of Grassmannians of two-planes
本文给出了格拉斯曼簇Gr(2,n)的Plücker坐标环的Frobenius重数矩阵的非递归公式，证明其元素为p的多项式，从而得到F-签名等为p的有理函数，并给出Gr(2,4)等的具体表达式。
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