arXiv subjects
Publications and source records attributed to Nathaniel Gallup.
We show that the (non-Noetherian) Stanley-Reisner ring of the order complex of certain intervals in the Bruhat order on the infinite symmetric group $S_\infty$ of all auto-bijections of $\mathbb{N}$ is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. This gives an infinite-dimensional version of results due to Edelman, Bj\"{o}rner, and Kind and Kleinschmidt for finite symmetric groups $S_n$.
We show that, under certain constraints, the Stanley-Reisner ring of an infinite simplicial complex is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. We apply this result to prove the wanted claim -- that initial complexes of matrix Schubert varieties corresponding to infinite permutations in $S_{\infty}$ with respect to an antidiagonal term order are Cohen-Macaulay (in the same sense), giving rise to new examples of non-Noetherian Cohen-Macaulay rings.
We prove a version of Gabriel's theorem for locally finite-dimensional representations of infinite quivers. Specifically, we show that if $\Omega$ is any connected quiver, the category of locally finite-dimensional representations of $\Omega$ has unique representation type (meaning no two indecomposable representations have the same dimension vector) if and only if the underlying graph of $\Omega$ is a generalized ADE Dynkin diagram (i.e. one of $A_n, D_n, E_6, E_7, E_8, A_{\infty}, A_{\infty , \infty}$ or $D_\infty$). This result is companion to earlier work of the authors generalizing Gabriel's theorem to infinite quivers with different conditions.
We resolve Stillman's conjecture for families of polynomial rings that are graded by any semigroup under mild conditions. Conversely, we show that these conditions are necessary for the existence of a Stillman bound. This has applications even for the well-known standard graded case.
We prove a version of Gabriel's theorem for (possibly infinite dimensional) representations of infinite quivers. More precisely, we show that the representation theory of quiver $\Omega$ is of unique type (each dimension vector has at most one associated indecomposable) and infinite Krull-Schmidt (every, possibly infinite dimensional, representation is a direct sum of indecomposables) if and only if $\Omega$ is eventually outward and of generalized ADE Dynkin type ($A_n$, $D_n$, $E_6$, $E_7$, $E_8$, $A_\infty$, $A_{\infty, \infty}$, or $D_\infty$). Furthermore we define an analog of the Euler-Tits form on the space of eventually constant infinite roots and show that a quiver is of generalized ADE Dynkin type if and only if this form is positive definite. In this case the indecomposables are all locally finite-dimensional and eventually constant and correspond bijectively to the positive roots (i.e. those of length $1$).
With the goal of computing the Grothendieck group of certain multigraded infinite polynomial rings and the $K$-series of infinite matrix Schubert spaces, we introduce a new type of $\Gamma$-graded $k$-algebra (which we call a PDCF algebra) and a new type of graded module (a BDF module) over said algebra. Since infinite polynomial rings are not Noetherian and the modules of interest are not finitely generated, we compensate by requiring certain finiteness properties of the grading group $\Gamma$ and of the grading itself. If $R$ is a PDCF $\Gamma$-graded $k$-algebra, we prove that every projective BDF $\Gamma$-graded $R$-module is free, and that when the graded maximal ideal of $R$ is generated by a regular sequence, a BDF analog of the Hilbert Syzygy Theorem holds: every BDF $R$-module has a free resolution of BDF $R$-modules which, though not finite, has the property that each graded piece is eventually zero. We use this to show that the Grothendieck group of projective BDF $R$-modules (defined similarly to $K_0(R)$) is isomorphic to the Grothendieck group of all BDF $R$-modules (defined similarly to $G_0(R)$). We describe this Grothendieck group explicitly by using $K$-series to give an isomorphism with a certain space of formal Laurent series. Finally we give a BDF version of Serre's formula for the product in the Grothendieck group, making it into a ring.
Using linear algebraic methods we show that every (possibly infinite-dimensional) representation of a quiver with underlying graph $A_{\infty, \infty}$ is Krull-Schmidt, as long as the arrows in the quiver eventually point outward.
Using Grothendieck's "functor of points" approach to algebraic geometry, we define a new infinite-dimensional algebro-geometric flag space as a $k$-functor (for $k$ a ring) which maps a $k$-algebra $R$ to the set of certain well-ordered chains of submodules of an infinite rank free $R$-module. This generalizes the well known construction of a $k$-functor that is represented by the classical (i.e. finite-dimensional) full flag scheme. We prove that as in the finite-dimensional case, there is an action of a general linear group on our flag space, that the stabilizer of the standard flag is the subgroup $B$ of upper triangular matrices, and that the Bruhat decomposition holds, meaning that our space is covered by the disjoint Schubert cells $\text{sh}(B σB) / B$ indexed by permutations $σ$ of an infinite set. Finally, in the case of flags indexed by the ordinal $ω+ 1$, we define an analog of the Bruhat order on this infinite permutation group and prove that when $k$ is a domain, Ehresmann's closure relations still hold, i.e. that the closure $\overline{\text{sh}(B σB) / B}$ is covered by the Schubert cells indexed by permutations smaller than $σ$ in the infinite Bruhat order.
We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces of algebraic curves in $B^4 \subset \mathbb{C}^2$, is applied to find contact 3-manifolds with topologically distinct symplectic fillings, and is generalized to higher dimensions.