Research
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extended abstract (submitted)
For each Coxeter element c in the symmetric group, we define a pattern- avoiding Birkhoff subpolytope whose vertices are the c-singletons. We show that the normalized volume of our polytope is equal to the number of longest chains in a corresponding type A Cambrian lattice. Our work extends a result of Davis and Sagan which states that the normalized volume of the convex hull of the 132 and 312 avoiding permutation matrices is the number of longest chains in the Tamari lattice, a special case of a type A Cambrian lattice. Furthermore, we prove that each of our polytopes is unimodularly equivalent to the order polytope of the heap of the c-sorting word of the longest permutation. This gives an affirmative answer to a generalization of a question posed by Davis and Sagan.
[7] Top-degree components of Grothendieck and Lascoux polynomials (with T. Yu), to appear in Algebraic Combinatorics
The Castelnuovo–Mumford polynomial $\widehat{\mathfrak{G}}_w$ with $w \in S_n$ is the highest homogeneous component of the Grothendieck polynomial $\mathfrak{G}_w$. Pechenik, Speyer and Weigandt define a statistic $\rajcode(\cdot)$ on $S_n$ that gives the leading monomial of $\widehat{\mathfrak{G}}_w$. We introduce a statistic rajcode on any diagram $D$ through a combinatorial construction ``snow diagram'' that augments and decorates $D$. When $D$ is the Rothe diagram of a permutation w, rajcode(D) agrees with the aforementioned $\rajcode(w)$. When D is the key diagram of a weak composition $\alpha$, $\rajcode(D)$ yields the leading monomialof $\widehat{\mathfrak{L}}_\alpha$, the highest homogeneous component of the Lascoux polynomials $\mathfrak{L}_\alpha$. We use $\widehat{\mathfrak{L}}_\alpha$ to construct a basis of $\widehat{V}_n$, the span of $\widehat{\mathfrak{G}}_w$ with $w \in S_n$. Then we show $\widehat{V}_n$ gives a natural algebraic interpretation of a classical q-analogue of Bell numbers.
[6] RSK tableaux and the weak order on fully commutative permutations (with E. Gunawan, H. Russell and B. Tenner), The Electronic Journal of Combinatorics 30(4) (2023)
For each fully commutative permutation, we construct a "boolean core," which is the maximal boolean permutation in its principal order ideal under the right weak order. We partition the set of fully commutative permutations into the recently defined crowded and uncrowded elements, distinguished by whether or not their RSK insertion tableaux satisfy a sparsity condition. We show that a fully commutative element is uncrowded exactly when it shares the RSK insertion tableau with its boolean core. We present the dynamics of the right weak order on fully commutative permutations, with particular interest in when they change from uncrowded to crowded. In particular, we use consecutive permutation patterns and descents to characterize the minimal crowded elements under the right weak order.
[5] Runs and RSK tableaux of boolean permutations (with E. Gunawan, H. Russell and B. Tenner), to appear in Annuals of Combinatorics
We define and construct the "canonical reduced word" of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the "run" statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean.
[4] A bijection between K-Kohnert diagrams and reverse set-valued tableaux (with T. Yu), The Electronic Journal of Combinatorics 30(4) (2023)
Lascoux polynomials are K-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux (𝖱𝖲𝖵𝖳) rule for Lascoux polynomials and reverse semistandard Young tableaux (𝖱𝖲𝖲𝖸𝖳) rule for key polynomials. Furthermore, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with 𝖱𝖲𝖲𝖸𝖳. Ross and Yong introduced K-Kohnert diagrams, which are analogues of Kohnert diagrams. They conjectured a K-Kohnert diagram rule for Lascoux polynomials. We establish this conjecture by constructing a weight-preserving bijection between 𝖱𝖲𝖵𝖳 and K-Kohnert diagrams.
[3] Uncrowding algorithm on hook-valued tableaux (with J. Pappe, W. Poh and A. Schilling), Annals of Combinatorics 26, 261-301 (2022)
Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm "uncrowds" the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that "uncrowds" the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials.
[2] A crystal on decreasing factorizations in the 0-Hecke monoid (with J. Morse, W. Poh and A. Schilling), The Electronic Journal of Combinatorics 27(2) (2020)
FPSAC 2020 extended abstract. arXiv:1911.08732. slides poster
We introduce a type A crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call ⋆-crystal. This crystal is a K-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the ⋆-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.
[1] Virtualization map for the Littelmann path model (with T. Scrimshaw), Transformation Groups (2018) 23: 1045
We show the natural embedding of weight lattices from a diagram folding is a virtualization map for the Littelmann path model, which recovers a result of Kashiwara. As an application, we give a type independent proof that certain Kirillov--Reshetikhin crystals respect diagram foldings, which is a known result on a special case of a conjecture given by Okado, Schilling, and Shimozono.