Skip to main content
eScholarship
Open Access Publications from the University of California

UC Davis is situated in the heart of California. Founded in 1933, the Department of Mathematics stays relevant in mathmatics research both through continuing research and in discovering new talent. Research is encouraged across all levels, from undergraduate through graduate, as well as through outreach programs.

Cover page of An insertion algorithm on multiset partitions with applications to diagram algebras

An insertion algorithm on multiset partitions with applications to diagram algebras

(2020)

© 2020 Elsevier Inc. We generalize the Robinson–Schensted–Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length k with entries in [n] and pairs of tableaux of the same shape with one being a standard Young tableau of size n and the other being a standard multiset tableau of content [k]. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson [15].

Cover page of A Liquid Crystal Model of Viral DNA Encapsidation

A Liquid Crystal Model of Viral DNA Encapsidation

(2020)

A liquid crystal continuum modeling framework for icosahedra bacteriophage viruses is developed and tested. The main assumptions of the model are the chromonic columnar hexagonal structure of confined DNA, the high resistance to bending and the phase transition from solid to fluid-like states as the concentration of DNA in the capsid decreases during infection. The model predicts osmotic pressure inside the capsid and the ejection force of the DNA as well as the size of the isotropic volume at the center of the capsid. Extensions of the model are discussed.

Cover page of Holonomy theorem for finite semigroups

Holonomy theorem for finite semigroups

(2020)

We provide a simple proof of the Holonomy Theorem using a new Lyndon-Chiswell length function on the Karnofsky-Rhodes expansion of a semigroup. Unexpectedly, we have both a left and a right action on the Chiswell tree by elliptic maps.

Cover page of The forgotten monoid

The forgotten monoid

(2020)

We study properties of the forgotten monoid which appeared in work of Lascoux and Schutzenberger and recently resurfaced in the construction of dual equivalence graphs by Assaf. In particular, we provide an explicit characterization of the forgotten classes in terms of inversion numbers and show that there are n^2-3n+4 forgotten classes in the symmetric group S_n. Each forgotten class contains a canonical element that can be characterized by pattern avoidance. We also show that the sum of Gessel's quasi-symmetric functions over a forgotten class is a 0-1 sum of ribbon-Schur functions.

Cover page of Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function

Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function

(2020)

It has previously been shown that, at least for non-exceptional Kac-Moody Lie algebras, there is a close connection between Demazure crystals and tensor products of Kirillov-Reshetikhin crystals. In particular, certain Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov-Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to Macdonald polynomials and q-deformed Whittaker functions.

Cover page of Quantum Lakshmibai-Seshadri paths and root operators

Quantum Lakshmibai-Seshadri paths and root operators

(2020)

We give an explicit description of the image of a quantum LS path, regarded as a rational path, under the action of root operators, and show that the set of quantum LS paths is stable under the action of the root operators. As a by-product, we obtain a new proof of the fact that a projected level-zero LS path is just a quantum LS path.

Cover page of Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths

Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths

(2020)

We give an explicit and computable description, in terms of the parabolic quantum Bruhat graph, of the degree function defined for quantum Lakshmibai-Seshadri paths, or equivalently, for "projected" (affine) level-zero Lakshmibai-Seshadri paths. This, in turn, gives an explicit and computable description of the global energy function on tensor products of Kirillov-Reshetikhin crystals of one-column type, and also of (classically restricted) one-dimensional sums.

Cover page of Markov chains through semigroup graph expansions (a survey)

Markov chains through semigroup graph expansions (a survey)

(2020)

We review the recent approach to Markov chains using the Karnofksy-Rhodes and McCammond expansions in semigroup theory by the authors and illustrate them by two examples.

Cover page of A uniform model for Kirillov-Reshetikhin crystals. Extended abstract

A uniform model for Kirillov-Reshetikhin crystals. Extended abstract

(2020)

We present a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai-Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction leads to a simple calculation of the energy function. It also implies the equality between a Macdonald polynomial specialized at t=0 and the graded character of a tensor product of KR modules.

Cover page of Recursions for rational q,t-Catalan numbers

Recursions for rational q,t-Catalan numbers

(2020)

© 2020 Elsevier Inc. We give a simple recursion labeled by binary sequences which computes rational q,t-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between the Khovanov-Rozansky homology of N,M-torus links and the rational q,t-Catalan power series for general positive N,M.