scholarly journals ON THE SYMMETRIC SQUARES OF COMPLEX AND QUATERNIONIC PROJECTIVE SPACE

2018 ◽  
Vol 60 (3) ◽  
pp. 703-729 ◽  
Author(s):  
YUMI BOOTE ◽  
NIGEL RAY
Keyword(s):  
Cohomology Ring ◽  
Theoretical Physics ◽  
Configuration Spaces ◽  
Quaternionic Projective Space ◽  
Fibonacci Polynomials ◽  
Cohomology Rings ◽  
Generators And Relations ◽  
Thom Spaces ◽  
Symmetric Square ◽  
The One

AbstractThe problem of computing the integral cohomology ring of the symmetric square of a topological space has long been of interest, but limited progress has been made on the general case until recently. We offer a solution for the complex and quaternionic projective spaces$\mathbb{K}$Pn, by utilising their rich geometrical structure. Our description involves generators and relations, and our methods entail ideas from the literature of quantum chemistry, theoretical physics, and combinatorics. We begin with the case$\mathbb{K}$P∞, and then identify the truncation required for passage to finiten. The calculations rely upon a ladder of long exact cohomology sequences, which compares cofibrations associated to the diagonals of the symmetric square and the corresponding Borel construction. These incorporate the one-point compactifications of classic configuration spaces of unordered pairs of points in$\mathbb{K}$Pn, which are identified as Thom spaces by combining Löwdin's symmetric orthogonalisation (and its quaternionic analogue) with a dash ofPingeometry. The relations in the ensuing cohomology rings are conveniently expressed using generalised Fibonacci polynomials. Our conclusions are compatible with those of Gugnin mod torsion and Nakaoka mod 2, and with homological results of Milgram.

scholarly journals The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A

2017 ◽  
Vol 2019 (17) ◽  
pp. 5316-5388 ◽  
Author(s):  
Hiraku Abe ◽  
Megumi Harada ◽  
Tatsuya Horiguchi ◽  
Mikiya Masuda
Keyword(s):  
Commutative Algebra ◽  
Cohomology Ring ◽  
Regular Sequence ◽  
Cohomology Rings ◽  
Generators And Relations ◽  
Hessenberg Varieties ◽  
Special Cases ◽  
Hessenberg Variety ◽  
Fixed Positive Integer ◽  
Special Case

AbstractLet $n$ be a fixed positive integer and $h: \{1,2,\ldots,n\} \rightarrow \{1,2,\ldots,n\}$ a Hessenberg function. The main results of this paper are two-fold. First, we give a systematic method, depending in a simple manner on the Hessenberg function $h$, for producing an explicit presentation by generators and relations of the cohomology ring $H^\ast({\mathrm{Hess}}(\mathsf{N},h))$ with ${\mathbb Q}$ coefficients of the corresponding regular nilpotent Hessenberg variety ${\mathrm{Hess}}(\mathsf{N},h)$. Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring $H^*({\mathrm{Hess}}(\mathsf{N},h))$ of the regular nilpotent Hessenberg variety and the $\mathfrak{S}_n$-invariant subring $H^*({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n}$ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the $\mathfrak{S}_n$-action on $H^*({\mathrm{Hess}}(\mathsf{S},h))$ defined by Tymoczko). Our second main result implies that $\mathrm{dim}_{{\mathbb Q}} H^k({\mathrm{Hess}}(\mathsf{N},h)) = \mathrm{dim}_{{\mathbb Q}} H^k({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n}$ for all $k$ and hence partially proves the Shareshian–Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley–Stembridge conjecture. A proof of the full Shareshian–Wachs conjecture was recently given by Brosnan and Chow, and independently by Guay–Paquet, but in our special case, our methods yield a stronger result (i.e., an isomorphism of rings) by more elementary considerations. This article provides detailed proofs of results we recorded previously in a research announcement [2].

scholarly journals Schubert Calculus on a Grassmann Algebra

2009 ◽  
Vol 52 (2) ◽  
pp. 200-212 ◽  
Author(s):  
Letterio Gatto ◽  
Taíse Santiago
Keyword(s):  
Cohomology Ring ◽  
Grassmann Algebra ◽  
Monic Polynomial ◽  
Free Module ◽  
Exterior Algebra ◽  
Schubert Calculus ◽  
Cohomology Rings ◽  
Generators And Relations ◽  
Small Quantum ◽  
Factorization Algebra

AbstractThe (classical, small quantum, equivariant) cohomology ring of the grassmannian G(k, n) is generated by certain derivations operating on an exterior algebra of a free module of rank n (Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree n into the product of two monic polynomials, one of degree k.

A solution of the time paradox of physics

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Grit Kalies
Keyword(s):  
Laws Of Nature ◽  
Theoretical Physics ◽  
Quantum Level ◽  
Energy Equivalence ◽  
Time Arrow ◽  
Laws Of Thermodynamics ◽  
Time Concepts ◽  
Modern Physics ◽  
The One ◽  
Matter Energy

AbstractQuantum mechanics for describing the behavior of microscopic entities and thermodynamics for describing macroscopic systems exhibit separate time concepts. Whereas many theories of modern physics interpret processes as reversible, in thermodynamics, an expression for irreversibility and the so-called time arrow has been developed: the increase of entropy. The divergence between complete reversibility on the one hand and irreversibility on the other is called the paradox of time. Since more than hundred years many efforts have been devoted to unify the time concepts. So far, the efforts were not successful. In this paper a solution is proposed on the basis of matter-energy equivalence with an energetic distinction between matter and mass. By refraining from interpretations predominant in modern theoretical physics, the first and second laws of thermodynamics can be extended to fundamental laws of nature, which are also valid at quantum level.

LORENTZ COVARIANCE FOR THE QUANTUM ALGEBRA OF OBSERVABLES: NAMBU–GOTO STRINGS IN 3 + 1 DIMENSIONS

2002 ◽  
Vol 17 (17) ◽  
pp. 2331-2349 ◽  
Author(s):  
GERRIT HANDRICH
Keyword(s):  
Rest Frame ◽  
Poisson Algebra ◽  
Quantum Algebra ◽  
Substantial Part ◽  
Quantum Case ◽  
Generators And Relations ◽  
Defining Relations ◽  
Algebra Of Observables ◽  
The One ◽  
The Relationship

To postulate correspondence for the observables only is a promising approach to a fully satisfying quantization of the Nambu–Goto string. The relationship between the Poisson algebra of observables and the corresponding quantum algebra is established in the language of generators and relations. A very valuable tool is the transformation to the string's rest frame, since a substantial part of the relations are solved. It is the aim of this paper to clarify the relationship between the fully covariant and the rest frame description. Both in the classical and in the quantum case, an efficient method for recovering the covariant algebra from the one in the rest frame is described. Restrictions on the quantum defining relations are obtained, which are not taken into account when one postulates correspondence for the rest frame algebra. For the part of the algebra studied up to now in explicit computations, these further restrictions alone determine the quantum algebra uniquely — in full consistency with the further restrictions found in the rest frame.

scholarly journals Cohomology of groups and splendid equivalences of derived categories

2001 ◽  
Vol 131 (3) ◽  
pp. 459-472 ◽  
Author(s):  
ALEXANDER ZIMMERMANN
Keyword(s):  
Group Ring ◽  
Local Structure ◽  
Cohomology Ring ◽  
Derived Categories ◽  
Derived Category ◽  
Restriction Map ◽  
Group Rings ◽  
Cohomology Rings ◽  
The Impact ◽  
Cohomology Of Groups

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

scholarly journals Was Pierre Duhem an Esprit de finesse?

2017 ◽  
pp. 93
Author(s):  
Víctor Manuel Hernández
Keyword(s):  
Physical Chemistry ◽  
Theoretical Physics ◽  
English School ◽  
Psychological Profile ◽  
Psychological Profiles ◽  
Cognitive Value ◽  
Pierre Duhem ◽  
The One ◽  
Crucial Experiments ◽  
The Ideal

Although Pierre Duhem is well known for his conventionalist outlook and, in particular, for his critique of crucial experiments outlined in his thesis on the empirical indeterminacy of theory, he also contributed to the scholarship on the psychological profiles of scientists by revising Pascal’s famous distinction between the subtle mind and the geometric mind (esprits fins and esprits géométriques). For Duhem, the ideal scientist is the one who combines the defining qualities of both types of intellect. As a physicist, Duhem made important theoretical contributions to the field of thermodynamics as well as to the then-nascent physical chemistry. Due to his rejection of atomism and his unrelenting critique of Maxwell’s electrodynamics, however, in his later years, Duhem’s work was surpassed and abandoned by the dominant tendencies of physics of the time. In this essay, I will discuss whether Duhem himself can be understood through the lens of his own account of the scientist’s psychological profile. More specifically, I examine whether the subtle mind – to which he seems to assign greater cognitive value – in fact plays a key role in Duhem’s critique of the English School (école anglaise), or if his preference for the axiomatic structure of theoretical physics shows a greater affinity with the geometric mind.

scholarly journals Double theta polynomials and equivariant Giambelli formulas

2015 ◽  
Vol 160 (2) ◽  
pp. 353-377 ◽  
Author(s):  
HARRY TAMVAKIS ◽  
ELIZABETH WILSON
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Generators And Relations ◽  
Raising Operators ◽  
Schubert Classes ◽  
Symplectic Grassmannians

AbstractWe use Young's raising operators to introduce and study double theta polynomials, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur S-polynomials and Q-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations.

The varieties of the mod p cohomology rings of extra special p-groups for an odd prime p

1983 ◽  
Vol 94 (3) ◽  
pp. 449-459 ◽  
Author(s):  
Michishige Tezuka ◽  
Nobuaki Yagita
Keyword(s):  
Cohomology Ring ◽  
Cohomology Rings ◽  
Mod P

In this paper we attempt to determine the mod p cohomology rings of the extra special p-groups for an odd prime p. Quillen has calculated the mod 2 case [8]. In our case, its structure seems to be very complicated [6]. It seems reasonable that we consider the variety defined by the cohomology ring of even degree according to Serre [11]. The main result of this paper (Theorem 5.3) gives a subalgebra of the cohomology ring such that the inclusion homomorphism induces a homeomorphism between the varieties.

scholarly journals Projective bundles over toric surfaces

2016 ◽  
Vol 27 (04) ◽  
pp. 1650032 ◽  
Author(s):  
Suyoung Choi ◽  
Seonjeong Park
Keyword(s):  
Toric Variety ◽  
Cohomology Ring ◽  
Line Bundles ◽  
Projective Bundle ◽  
Cohomology Rings ◽  
Toric Surfaces ◽  
Projective Bundles ◽  
Higher Dimensional ◽  
Quasitoric Manifolds

Let [Formula: see text] be the Whitney sum of complex line bundles over a topological space [Formula: see text]. Then, the projectivization [Formula: see text] of [Formula: see text] is called a projective bundle over [Formula: see text]. If [Formula: see text] is a nonsingular complete toric variety, then so is [Formula: see text]. In this paper, we show that the cohomology ring of a nonsingular projective toric variety [Formula: see text] determines whether it admits a projective bundle structure over a nonsingular complete toric surface. In addition, we show that two [Formula: see text]-dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds are diffeomorphic if their cohomology rings are isomorphic as graded rings. Furthermore, we study the smooth classification of higher dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds.

scholarly journals The cohomology ring of the sapphires that admit the Sol geometry

2018 ◽  
Vol 28 (03) ◽  
pp. 365-380 ◽  
Author(s):  
Daciberg Lima Gonçalves ◽  
Sérgio Tadao Martins
Keyword(s):  
Fundamental Group ◽  
Cohomology Ring ◽  
Free Resolution ◽  
Multiplicative Structure ◽  
Torus Bundle ◽  
Cohomology Rings ◽  
Cup Product ◽  
Diagonal Approximation

Let [Formula: see text] be the fundamental group of a sapphire that admits the Sol geometry and is not a torus bundle. We determine a finite free resolution of [Formula: see text] over [Formula: see text] and calculate a partial diagonal approximation for this resolution. We also compute the cohomology rings [Formula: see text] for [Formula: see text] and [Formula: see text] for an odd prime [Formula: see text], and indicate how to compute the groups [Formula: see text] and the multiplicative structure given by the cup product for any system of coefficients [Formula: see text].

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