53 research outputs found
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 when 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 be a graded poset of rank and let be a -element
chain. For an order ideal of , its rowmotion
is the smallest ideal containing the minimal elements of the complementary
filter of . The map defines invertible dynamics on the set of ideals.
We say that that has NRP ("not relatively prime") rowmotion if no
-orbit has cardinality relatively prime to .
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 of two chains, the poset whose order ideals correspond to
the Schubert varieties of a Grassmann variety
under containment. Here, we initiate the general study of posets with NRP
rowmotion.
Our first main result establishes NRP rowmotion for all minuscule posets ,
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 .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
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 . In recent work, we
used this viewpoint to develop topologically-motivated bases of
and initiate a Schubert calculus for in both
cohomology and -theory.
Here, we study the torus-equivariant cohomology of
. 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
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