Singular nonsymmetric Jack polynomials for some trapeziform tableaux

2021 ◽  
pp. 2150091
Author(s):  
Juanjuan Sun ◽  
Shiru Wu
Keyword(s):  
Symmetric Group ◽  
Jack Polynomials ◽  
Type Formula

Let [Formula: see text] be symmetric group of degree [Formula: see text]. In the representation of [Formula: see text] corresponding to trapeziform partition (precisely [Formula: see text], [Formula: see text]), we propose nonsymmetric Jack polynomials in [Formula: see text] variables which are singular at [Formula: see text]. Moreover those nonsymmetric Jack polynomials span a module of [Formula: see text] which is isomorphic to the representation of type [Formula: see text].

Galois groups of Taylor polynomials of some elementary functions

2019 ◽  
Vol 15 (06) ◽  
pp. 1127-1141
Author(s):  
Khosro Monsef Shokri ◽  
Jafar Shaffaf ◽  
Reza Taleb
Keyword(s):  
Symmetric Group ◽  
Galois Groups ◽  
Alternating Group ◽  
Integer Number ◽  
Elementary Functions ◽  
Type Formula ◽  
Number Formula ◽  
Taylor Polynomials ◽  
Index 2

Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions [Formula: see text] and [Formula: see text]. We first show that the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] are as large as possible, namely, [Formula: see text] (full symmetric group) or [Formula: see text] (alternating group), depending on the residue of the integer number [Formula: see text] modulo [Formula: see text]. We then compute the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] and show that these Galois groups essentially coincide with the Coexter groups of type [Formula: see text] (or an index 2 subgroup of the corresponding Coexter group).

scholarly journals Verlinde formulae on complex surfaces: K-theoretic invariants

2021 ◽  
Vol 9 ◽  
Author(s):  
L. Göttsche ◽  
M. Kool ◽  
R. A. Williams
Keyword(s):  
Moduli Space ◽  
K3 Surfaces ◽  
Complex Surfaces ◽  
Type Formula ◽  
Verlinde Formula ◽  
Donaldson Invariants

Abstract We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).

The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

2020 ◽  
Author(s):  
Heather M Russell ◽  
Julianna Tymoczko
Keyword(s):  
Symmetric Group ◽  
Partial Order ◽  
Transition Matrix ◽  
Group Representation ◽  
Jones Polynomial ◽  
Basis Matrix ◽  
Combinatorial Structures ◽  
Quantum Invariants ◽  
One Dimensional ◽  
Link Diagrams

Abstract Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, which measure depths of regions inside the web. As an application, we resolve an open conjecture that the change of basis between the so-called Specht basis and web basis of this symmetric group representation is unitriangular for $\mathfrak{sl}_3$-webs ([ 33] and [ 29].) We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.

scholarly journals A Class of Integral Operators that Fix Exponential Functions

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Carlo Bardaro ◽  
Ilaria Mantellini ◽  
Gumrah Uysal ◽  
Basar Yilmaz
Keyword(s):  
General Class ◽  
Pointwise Convergence ◽  
Type Theorem ◽  
Integral Operators ◽  
Exponential Functions ◽  
Convergence Theorems ◽  
Korovkin Type Theorem ◽  
Type Formula

AbstractIn this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

SYMMETRIC GROUP BLOCKS OF DEFECT TWO

1995 ◽  
Vol 46 (2) ◽  
pp. 201-234 ◽  
Author(s):  
JOANNA SCOPES
Keyword(s):  
Symmetric Group

scholarly journals Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

2021 ◽  
Author(s):  
Andreas Bernig ◽  
Dmitry Faifman ◽  
Gil Solanes
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Riemannian Space ◽  
Space Forms ◽  
Type Formula ◽  
Isometric Embeddings ◽  
Curvature Measures ◽  
Riemannian Space Forms

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

Electron localisation, Lyapunov exponents and the symmetric group

1989 ◽  
Vol 1 (19) ◽  
pp. 3073-3082 ◽  
Author(s):  
K Slevin ◽  
E Castano ◽  
J B Pendry
Keyword(s):  
Symmetric Group ◽  
Lyapunov Exponents

scholarly journals The Round Functions of Cryptosystem PGM Generate the Symmetric Group

2006 ◽  
Vol 38 (1) ◽  
pp. 147-155 ◽  
Author(s):  
A. Caranti ◽  
F. Dalla. Volta
Keyword(s):  
Symmetric Group
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