论文标题
确定增厚的共同体
Syzygies of Determinantal Thickenings
论文作者
论文摘要
令$ s = \ mathbb {c} [x_ {i，j}] $为$ m \ times n $矩阵的多项式函数的环，并考虑组$ \ mathbf {gl} = \ \ \ \ m m mathbf {gl} {gl} {gl} _mm \ times \ times \ \ \ \ \ \ \ \ mathbf {gl gl} $ on protr and the matr and compia和compl the MATR和comptries the MATR和comply compia和complies the MATR和comptiation。 Raicu and Weyman证明，对于$ \ mathbf {gl} $ - 不变的理想$ i \ subseteq s $，其最小免费分辨率的线性线性通过BGG对应于模量的通用模词，上面是通用线性linear linear line superalgebra lie superalgebra $ \ mathfrak $ \ mathfrak {gl}（gl}（m | n）$。当$ i =i_λ$是由$ \ mathbf {gl} $ - 重量$λ$的最高权重向量产生的理想时，他们对Grothendieck Group中这些$ \ mathfrak {Gl}（M | n）$ - 模块的类别进行了猜想的描述。我们在这里证明了他们的猜想。我们还提供了有关如何获取这些$ \ mathfrak {gl}（m | n）$ - 模块的算法描述。
Let $S = \mathbb{C}[x_{i,j}]$ be the ring of polynomial functions on the space of $m \times n$ matrices, and consider the action of the group $\mathbf{GL} = \mathbf{GL}_m \times \mathbf{GL}_n$ via row and column operations on the matrix entries. It is proven by Raicu and Weyman that for a $\mathbf{GL}$-invariant ideal $I \subseteq S$, the linear strands of its minimal free resolution translates via the BGG correspondence to modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. When $I=I_λ$ is the ideal generated by the $\mathbf{GL}$-orbit of a highest weight vector of weight $λ$, they gave a conjectural description of the classes of these $\mathfrak{gl}(m|n)$-modules in the Grothendieck group. We prove their conjecture here. We also give a algorithmic description of how to get the classes of these $\mathfrak{gl}(m|n)$-modules for any $\mathbf{GL}$-invariant ideal $I \subseteq S$.
