Degenerate flag varieties: moment graphs and Schr\"oder numbers

We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schr\"oder number and the Poincar\'e polynomial is given by a natural statistics counting the number of diagonal steps in a Schr\"oder path. As an application we obtain a new combinatorial description of the large and small Schr\"oder numbers and their q-analogues.Comment: 25 page

ErbB receptors and cell polarity: New pathways and paradigms for understanding cell migration and invasion

The ErbB family of receptor tyrosine kinases is involved in initiation and progression of a number of human cancers, and receptor activation or overexpression correlates with poor patient survival. Research over the past two decades has elucidated the molecular mechanisms underlying ErbB-induced tumorigenesis, which has resulted in the development of effective targeted therapies. ErbB-induced signal transduction cascades regulate a wide variety of cell processes, including cell proliferation, apoptosis, cell polarity, migration and invasion. Within tumors, disruption of these core processes, through cooperative oncogenic lesions, results in aggressive, metastatic disease. This review will focus on the ErbB signaling networks that regulate migration and invasion and identify a potential role for cell polarity pathways during cancer progression

Free-Field Realization of D-dimensional Cylindrical Gravitational Waves

We find two-dimensional free-field variables for D-dimensional general relativity on spacetimes with D-2 commuting spacelike Killing vector fields and non-compact spatial sections for D>4. We show that there is a canonical transformation which maps the corresponding two-dimensional dilaton gravity theory into a two-dimensional diffeomorphism invariant theory of the free-field variables. We also show that the spacetime metric components can be expressed as asymptotic series in negative powers of the dilaton, with coefficients which can be determined in terms of the free fields.Comment: 15 pages, Late

Quantum Algebraic Approach to Refined Topological Vertex

We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W_{1+infty} introduced by Miki. Our construction involves trivalent intertwining operators Phi and Phi^* associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors in Z^2 is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of Phi and Phi^* give the refined topological vertex C_{lambda mu nu}(t,q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C_{lambda mu}^nu(q,t) of Awata-Kanno. The gluing factors appears correctly when we consider any compositions of Phi and Phi^*. The spectral parameters attached to Fock spaces play the role of the K"ahler parameters.Comment: 27 page

Free Boson Realization of Uq(slN^)U_q(\widehat{sl_N})

We construct a realization of the quantum affine algebra $U_q(\widehat{sl_N})$ of an arbitrary level $k$ in terms of free boson fields. In the $q\!\rightarrow\! 1$ limit this realization becomes the Wakimoto realization of $\widehat{sl_N}$. The screening currents and the vertex operators(primary fields) are also constructed; the former commutes with $U_q(\widehat{sl_N})$ modulo total difference, and the latter creates the $U_q(\widehat{sl_N})$ highest weight state from the vacuum state of the boson Fock space.Comment: 24 pages, LaTeX, RIMS-924, YITP/K-101

Generalized Calogero-Moser systems from rational Cherednik algebras

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero-Moser type which we explicitly specify. In the case of classical Coxeter groups we also obtain generalized Calogero-Moser systems with added quadratic potential.Comment: 36 pages; the main change is an improvement of section 7 so that it now deals with an arbitrary complex reflection group; Selecta Math, 201

G-protein-coupled receptor GPR161 is overexpressed in breast cancer and is a promoter of cell proliferation and invasion

Triple-negative breast cancer (TNBC) accounts for 20% of breast cancer in women and lacks an effective targeted therapy. Therefore, finding common vulnerabilities in these tumors represents an opportunity for more effective treatment. Despite the growing appreciation of G-protein-coupled receptor (GPCR)-mediated signaling in cancer pathogenesis, very little is known about the role GPCRs play in TNBC. Using genomic information of human breast cancer, we have discovered that the orphan GPCR, G-protein-coupled receptor 161 (GPR161) is overexpressed specifically in TNBC and correlates with poor prognosis. Knockdown of GPR161 impairs proliferation of human basal breast cancer cell lines. Overexpression of GPR161 in human mammary epithelial cells increases cell proliferation, migration, intracellular accumulation of E-cadherin, and formation of multiacinar structures in 3D culture. GPR161 forms a signaling complex with the scaffold proteins beta-arrestin 2 and Ile Gln motif containing GTPase Activating Protein 1, a regulator of mammalian target of rapamycin complex 1 and E-cadherin. Consistently, GPR161 amplified breast tumors and cells overexpressing GPR161 activate mammalian target of rapamycin signaling and decrease Ile Gln motif containing GTPase Activating Protein 1 phosphorylation. Thus, we identify the orphan GPCR, GPR161, as an important regulator and a potential drug target for TNBC

SLE local martingales in logarithmic representations

A space of local martingales of SLE type growth processes forms a representation of Virasoro algebra, but apart from a few simplest cases not much is known about this representation. The purpose of this article is to exhibit examples of representations where L_0 is not diagonalizable - a phenomenon characteristic of logarithmic conformal field theory. Furthermore, we observe that the local martingales bear a close relation with the fusion product of the boundary changing fields. Our examples reproduce first of all many familiar logarithmic representations at certain rational values of the central charge. In particular we discuss the case of SLE(kappa=6) describing the exploration path in critical percolation, and its relation with the question of operator content of the appropriate conformal field theory of zero central charge. In this case one encounters logarithms in a probabilistically transparent way, through conditioning on a crossing event. But we also observe that some quite natural SLE variants exhibit logarithmic behavior at all values of kappa, thus at all central charges and not only at specific rational values.Comment: 40 pages, 7 figures. v3: completely rewritten, new title, new result

A commutative algebra on degenerate CP^1 and Macdonald polynomials

We introduce a unital associative algebra A over degenerate CP^1. We show that A is a commutative algebra and whose Poincar'e series is given by the number of partitions. Thereby we can regard A as a smooth degeneration limit of the elliptic algebra introduced by one of the authors and Odesskii. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by one of the authors. It is found that the Ding-Iohara algebra provides us with an algebraic framework for the free filed construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting a la Babelon Bernard Billey, the Ruijsenaars difference operator and the operator M(q,t_1,t_2) of Okounkov-Pandharipande.Comment: 43 page

Factorizable ribbon quantum groups in logarithmic conformal field theories

We review the properties of quantum groups occurring as Kazhdan--Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.Comment: 27pp., amsart++, xy. v2: references added, some other minor addition