scholarly journals Representation Theory of Groups and D -Modules

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Ibrahim Nonkané
Keyword(s):  
Representation Theory ◽  
Polynomial Ring ◽  
Geometric Interpretation ◽  
Direct Image ◽  
Module Structure ◽  
D Modules

In this paper, we study a decomposition D -module structure of the polynomial ring. Then, we illustrate a geometric interpretation of the Specht polynomials. Using Brauer’s characterization, we give a partial generalization of the fact that factors of the discriminant of a finite map π : spec B ⟶ spec A generate the irreducible factors of the direct image of B under the map π .

scholarly journals Multivariate Alexander colorings

2018 ◽  
Vol 27 (14) ◽  
pp. 1850076 ◽  
Author(s):  
Lorenzo Traldi
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Knot Theory ◽  
Laurent Polynomial ◽  
Module Structure ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Linear Maps ◽  
Laurent Polynomial Ring ◽  
Special Case

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module [Formula: see text] over the Laurent polynomial ring [Formula: see text]. If [Formula: see text] is a diagram of a link [Formula: see text] with [Formula: see text] components, then the colorings of [Formula: see text] with values in [Formula: see text] form a [Formula: see text]-module [Formula: see text]. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J. 33 (2010) 116–122], we show that [Formula: see text] is isomorphic to the module of [Formula: see text]-linear maps from the Alexander module of [Formula: see text] to [Formula: see text]. In particular, suppose [Formula: see text] is a field and [Formula: see text] is a homomorphism of rings with unity. Then [Formula: see text] defines a [Formula: see text]-module structure on [Formula: see text], which we denote [Formula: see text]. We show that the dimension of [Formula: see text] as a vector space over [Formula: see text] is determined by the images under [Formula: see text] of the elementary ideals of [Formula: see text]. This result applies in the special case of Fox tricolorings, which correspond to [Formula: see text] and [Formula: see text]. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine [Formula: see text]; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications 10 (2001) 813–821; op. cit.].

Interpretation of generalized matrix functions via geometric measure of quantum entanglement

2015 ◽  
Vol 13 (07) ◽  
pp. 1550049
Author(s):  
Haixia Chang ◽  
Vehbi E. Paksoy ◽  
Fuzhen Zhang
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Quantum Entanglement ◽  
Geometric Interpretation ◽  
Matrix Functions ◽  
Irreducible Characters ◽  
Geometric Measure ◽  
Generalized Matrix

By using representation theory and irreducible characters of the symmetric group, we introduce character dependent states and study their entanglement via geometric measure. We also present a geometric interpretation of generalized matrix functions via this entanglement analysis.

scholarly journals $D$-modules and representation theory of Lie groups

1993 ◽  
Vol 43 (5) ◽  
pp. 1597-1618 ◽  
Author(s):  
Masaki Kashiwara
Keyword(s):  
Representation Theory ◽  
Lie Groups ◽  
D Modules

scholarly journals Representation theory of symmetric groups and the strong Lefschetz property

2020 ◽  
pp. 2250055
Author(s):  
Seok-Jin Kang ◽  
Young Rock Kim ◽  
Yong-Su Shin
Keyword(s):  
Representation Theory ◽  
Closed Form ◽  
Irreducible Component ◽  
Complete Intersection ◽  
Symmetric Groups ◽  
Module Structure ◽  
Structure And Properties ◽  
Lefschetz Property ◽  
Strong Lefschetz Property

We investigate the structure and properties of an Artinian monomial complete intersection quotient [Formula: see text]. We construct explicit homogeneous bases of [Formula: see text] that are compatible with the [Formula: see text]-module structure for [Formula: see text], all exponents [Formula: see text] and all homogeneous degrees [Formula: see text]. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible component appearing in the [Formula: see text]-module decomposition of homogeneous subspaces.

scholarly journals The Module Structure of a Group Action on a Polynomial Ring

1999 ◽  
Vol 218 (2) ◽  
pp. 672-692 ◽  
Author(s):  
Dikran B Karagueuzian ◽  
Peter Symonds
Keyword(s):  
Polynomial Ring ◽  
Group Action ◽  
Module Structure
Sign in / Sign up

Export Citation Format

Share Document