Crystal energy functions via the charge in types A and C

The Ram-Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara-Nakashima single column crystals. In this paper we prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial R-matrix.Comment: 25 pages; 1 figur

Stable Grothendieck polynomials and K-theoretic factor sequences

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of [Fomin-Greene '98] for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood-Richardson rule of [Buch '02]. The proof is based on a generalization of the Robinson-Schensted and Edelman-Greene insertion algorithms. Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of [Lascoux-Schutzenberger '82]. In particular, we provide the first $K$-theoretic analogue of the factor sequence formula of [Buch-Fulton '99] for the cohomological quiver polynomials

Separation of colour degree of freedom from dynamics in a soliton cellular automaton

We present an algorithm to reduce the coloured box-ball system, a one dimensional integrable cellular automaton described by motions of several colour (kind) of balls, into a simpler monochrome system. This algorithm extracts the colour degree of freedom of the automaton as a word which turns out to be a conserved quantity of this dynamical system. It is based on the theory of crystal basis and in particular on the tensor products of sl_n crystals of symmetric and anti-symmetric tensor representations.Comment: 19 page

Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A

We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out to be expressible only in terms of: - Schensted's classic bumping procedure, - the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to describe, acting on cylindric multipartitions. We explain how this can be seen as an analogue of the bumping algorithm for affine type $A$. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space

On Kazhdan-Lusztig cells in type B

32 pagesWe prove that, for any choice of parameters, the Kazhdan-Lusztig cells of a Weyl group of type $B$ are unions of combinatorial cells (defined using the domino insertion algorithm)

Nickel-Catalyzed Conversion of Enol Triflates into Alkenyl Halides

A Ni‐catalyzed halogenation of enol triflates was developed and it enables the synthesis of a broad range of alkenyl iodides, bromides, and chlorides under mild reaction conditions. The reaction utilizes inexpensive, bench‐stable Ni(OAc)_2⋅4 H_2O as a precatalyst and proceeds at room temperature in the presence of sub‐stoichiometric Zn and either 1,5‐cyclooctadiene or 4‐(N,N‐dimethylamino)pyridine

Cardiac-Specific SOCS3 Deletion Prevents In Vivo Myocardial Ischemia Reperfusion Injury through Sustained Activation of Cardioprotective Signaling Molecules.

Myocardial ischemia reperfusion injury (IRI) adversely affects cardiac performance and the prognosis of patients with acute myocardial infarction. Although myocardial signal transducer and activator of transcription (STAT) 3 is potently cardioprotective during IRI, the inhibitory mechanism responsible for its activation is largely unknown. The present study aimed to investigate the role of the myocardial suppressor of cytokine signaling (SOCS)-3, an intrinsic negative feedback regulator of the Janus kinase (JAK)-STAT signaling pathway, in the development of myocardial IRI. Myocardial IRI was induced in mice by ligating the left anterior descending coronary artery for 1 h, followed by different reperfusion times. One hour after reperfusion, the rapid expression of JAK-STAT-activating cytokines was observed. We precisely evaluated the phosphorylation of cardioprotective signaling molecules and the expression of SOCS3 during IRI and then induced myocardial IRI in wild-type and cardiac-specific SOCS3 knockout mice (SOCS3-CKO). The activation of STAT3, AKT, and ERK1/2 rapidly peaked and promptly decreased during IRI. This decrease correlated with the induction of SOCS3 expression up to 24 h after IRI in wild-type mice. The infarct size 24 h after reperfusion was significantly reduced in SOCS3-CKO compared with wild-type mice. In SOCS3-CKO mice, STAT3, AKT, and ERK1/2 phosphorylation was sustained, myocardial apoptosis was prevented, and the expression of anti-apoptotic Bcl-2 family member myeloid cell leukemia-1 (Mcl-1) was augmented. Cardiac-specific SOCS3 deletion led to the sustained activation of cardioprotective signaling molecules including and prevented myocardial apoptosis and injury during IRI. Our findings suggest that SOCS3 may represent a key factor that exacerbates the development of myocardial IRI

On the uniqueness of promotion operators on tensor products of type A crystals

The affine Dynkin diagram of type $A_n^{(1)}$ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type $A_n$ crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type $A_n$ crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara's conjecture that all `good' affine crystals are tensor products of Kirillov-Reshetikhin crystals.Comment: 31 pages; 8 figure