K-theoretic crystals for set-valued tableaux of rectangular shapes

In earlier work with C. Monical (2018), we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard set-valued tableaux of any fixed rectangular shape. Here, we establish this conjecture by explicitly constructing the K-crystal operators. As a consequence, we establish the first combinatorial formula for Lascoux polynomials $L_{w\lambda}$ when $\lambda$ is a multiple of a fundamental weight as the sum over flagged set-valued tableaux. Using this result, we then prove corresponding cases of conjectures of Ross-Yong (2015) and Monical (2016) by constructing bijections with the respective combinatorial objects.Comment: 20 pages, 2 figures; changed the statement of Conjecture 6.

Large Cardinals

Infinite sets are a fundamental object of modern mathematics. Surprisingly, the existence of infinite sets cannot be proven within mathematics. Their existence, or even the consistency of their possible existence, must be justified extra-mathematically or taken as an article of faith. We describe here several varieties of large infinite set that have a similar status in mathematics to that of infinite sets, i.e. their existence cannot be proven, but they seem both reasonable and useful. These large sets are known as large cardinals. We focus on two types of large cardinal: inaccessible cardinals and measurable cardinals. Assuming the existence of a measurable cardinal allows us to disprove a questionable statement known as the Axiom of Constructibility (V=L)

Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass

Let $P$ be a graded poset of rank $r$ and let $\mathbf{c}$ be a $c$-element chain. For an order ideal $I$ of $P \times \mathbf{c}$, its rowmotion $\psi(I)$ is the smallest ideal containing the minimal elements of the complementary filter of $I$. The map $\psi$ defines invertible dynamics on the set of ideals. We say that that $P$ has NRP ("not relatively prime") rowmotion if no $\psi$-orbit has cardinality relatively prime to $r+c+1$. In work with R. Patrias (2020), we proved a 1995 conjecture of P. Cameron and D. Fon-Der-Flaass by establishing NRP rowmotion for the product $P = \mathbf{a} \times \mathbf{b}$ of two chains, the poset whose order ideals correspond to the Schubert varieties of a Grassmann variety $\mathrm{Gr}_a(\mathbb{C}^{a+b})$ under containment. Here, we initiate the general study of posets with NRP rowmotion. Our first main result establishes NRP rowmotion for all minuscule posets $P$, posets whose order ideals reflect the Schubert stratification of minuscule flag varieties. Our second main result is that NRP promotion depends only on the isomorphism class of the comparability graph of $P$.Comment: 15 pages, 5 figure

James reduced product schemes and double quasisymmetric functions

Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that $\textrm{QSym}$ manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product $J\mathbb{C}\mathbb{P}^\infty$. In recent work, we used this viewpoint to develop topologically-motivated bases of $\textrm{QSym}$ and initiate a Schubert calculus for $J\mathbb{C}\mathbb{P}^\infty$ in both cohomology and $K$-theory. Here, we study the torus-equivariant cohomology of $J\mathbb{C}\mathbb{P}^\infty$. We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood--Richardson rule for the structure coefficients of this basis. Furthermore, we introduce an algebro-geometric analogue of the James reduced product construction. In particular, we prove that the James reduced product of a complex projective variety also carries the structure of a projective variety