The vertex operator for a generalization of MacMahon’s formula

2015 ◽  
Vol 30 (30) ◽  
pp. 1550176 ◽  
Author(s):  
Liqiang Cai ◽  
Lifang Wang ◽  
Ke Wu ◽  
Jie Yang
Keyword(s):  
Kernel Function ◽  
Vertex Operator ◽  
Symmetric Function ◽  
The State ◽  
Vertex Operators ◽  
Creation And Annihilation Operators ◽  
Operator Realization ◽  
Macdonald Symmetric Function ◽  
The Creation ◽  
Two Parameter

We provide a vertex operator realization for a two-parameter generalization of MacMahon’s formula introduced by M. Vuletić [Trans. Amer. Math. Soc. 361, 2789 (2009)]. Since the generalized MacMahon function is the kernel function of some Macdonald symmetric function, we consider the action of two vertex operators on a state corresponding to a Macdonald symmetric function. It becomes evident that the vertex operators appear to be the creation and annihilation operators, respectively on the state.

scholarly journals A Macdonald Vertex Operator and Standard Tableaux Statistics

10.37236/1383 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mike Zabrocki
Keyword(s):  
Vertex Operator ◽  
Generating Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Function ◽  
Inductive Proof ◽  
Integer Coefficients ◽  
The Family ◽  
The Relationship ◽  
Two Parameter

The two parameter family of coefficients $K_{\lambda \mu}(q,t)$ introduced by Macdonald are conjectured to $(q,t)$ count the standard tableaux of shape $\lambda $. If this conjecture is correct, then there exist statistics $a_\mu(T)$ and $b_\mu(T)$ such that the family of symmetric functions $H_\mu[X;q,t] = \sum_\lambda K_{\lambda \mu}(q,t) s_\lambda [X]$ are generating functions for the standard tableaux of size $|\mu|$ in the sense that $H_\mu[X;q,t] = \sum_{T} q^{a_\mu(T)} t^{b_\mu(T)} s_{\lambda (T)}[X]$ where the sum is over standard tableau of of size $|\mu|$. We give a formula for a symmetric function operator $H_2^{qt}$ with the property that $H_2^{qt} H_{(2^a1^b)}[X;q,t]= H_{(2^{a+1}1^b)}[X;q,t]$. This operator has a combinatorial action on the Schur function basis. We use this Schur function action to show by induction that $H_{(2^a1^b)}[X;q,t]$ is the generating function for standard tableaux of size $2a+b$ (and hence that $K_{\lambda (2^a1^b)}(q,t)$ is a polynomial with non-negative integer coefficients). The inductive proof gives an algorithm for 'building' the standard tableaux of size $n+2$ from the standard tableaux of size $n$ and divides the standard tableaux into classes that are generalizations of the catabolism type. We show that reversing this construction gives the statistics $a_\mu(T)$ and $b_\mu(T)$ when $\mu$ is of the form $(2^a1^b)$ and that these statistics prove conjectures about the relationship between adjacent rows of the $(q,t)$-Kostka matrix that were suggested by Lynne Butler.

scholarly journals Positivity for Special Cases of $(q,t)$-Kostka Coefficients and Standard Tableaux Statistics

10.37236/1473 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Mike Zabrocki
Keyword(s):  
Vertex Operator ◽  
Generating Functions ◽  
Symmetric Function ◽  
Vertex Operators ◽  
Kostka Polynomials ◽  
Special Cases

We present two symmetric function operators $H_3^{qt}$ and $H_4^{qt}$ that have the property $H_{m}^{qt} H_{(2^a1^b)}(X;q,t) = H_{(m2^a1^b)}(X;q,t)$. These operators are generalizations of the analogous operator $H_2^{qt}$ and also have expressions in terms of Hall-Littlewood vertex operators. We also discuss statistics, $a_{\mu}(T)$ and $b_{\mu}(T)$, on standard tableaux such that the $q,t$ Kostka polynomials are given by the sum over standard tableaux of shape $\lambda$, $K_{\lambda\mu}(q,t) = \sum_T t^{a_{\mu}(T)} q^{b_{\mu}(T)}$ for the case when when $\mu$ is two columns or of the form $(32^a1^b)$ or $(42^a1^b)$. This provides proof of the positivity of the $(q,t)$-Kostka coefficients in the previously unknown cases of $K_{\lambda (32^a1^b)}(q,t)$ and $K_{\lambda (42^a1^b)}(q,t)$. The vertex operator formulas are used to give formulas for generating functions for classes of standard tableaux that generalize the case when $\mu$ is two columns.

ACHIEVEMENTS AND PROSPECTS OF ONOMASTIC STANDARDIZATION: BELARUSIAN EXPERIENCE

2020 ◽  
pp. 72-82
Author(s):  
I.L. Kapylou
Keyword(s):  
Proper Names ◽  
Legal Framework ◽  
The State ◽  
The Creation

The article describes the achievements and determines the prospects for the standardization of Belarusian onyms: it examines the problems associated with the establishment of official written forms of toponyms, the creation of normative onomastic reference books, the functioning of onyms in the situation of the state Belarusian-Russian bilingualism in Belarus, the transliteration of foreign names into the Belarusian language, the preparation of a legal framework and development of a program for proper names romanization.

The Museum Istiqlal The Case of an Indonesian Islamic Museum

SUHUF ◽  
2015 ◽  
Vol 4 (2) ◽  
pp. 341-357
Author(s):  
Jonathan Zilberg
Keyword(s):  
Public Sphere ◽  
The State ◽  
Education Sector ◽  
Significant Potential ◽  
The Creation ◽  
The Government

This article describes the conflicted genesis of the Museum Istiqlal, the history of  the creation of the collection, and the state of the institution relative to other Indonesian museums. It emphasizes both  positive developments underway and the historical problems facing the institution. Above all, it focuses on the role the museum was originally intended to serve for the Indonesian Muslim public sphere and the significant potential the museum has to better serve that mission in the national and international sphere. In short, the article emphasizes that in the context of the Government of Indonesia’s current four year plan to revive the museum sector, the problems and opportunities presented at the Museum Istiqlal are symptomatic of endemic national challenges for both the museum and the education sector.

scholarly journals Relating the b ghost and the vertex operators of the pure spinor superstring

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Osvaldo Chandia ◽  
Brenno Carlini Vallilo
Keyword(s):  
Vertex Operator ◽  
Bosonic String ◽  
Vertex Operators ◽  
Pure Spinor ◽  
Double Pole ◽  
Lorenz Gauge

Abstract The OPE between the composite b ghost and the unintegrated vertex operator for massless states of the pure spinor superstring is computed and shown to reproduce the structure of the bosonic string result. The double pole vanishes in the Lorenz gauge and the single pole is shown to be equal to the corresponding integrated vertex operator.

Prologue to peacekeeping: Ireland and the Saar, 1934–5

1997 ◽  
Vol 30 (119) ◽  
pp. 420-428 ◽  
Author(s):  
Michael Kennedy
Keyword(s):  
League Of Nations ◽  
The State ◽  
Secretary General ◽  
Free State ◽  
International Labour Organisation ◽  
High Commissioner ◽  
The Status ◽  
Assistant Director ◽  
The Creation ◽  
International Peacekeeping

A list from September 1939 of files destroyed by the Department of External Affairs in the invasion scares of 1939–40 contains an intriguing reference to the possibility of dispatching Irish military forces to the Saarland on the Franco-German border in the winter of 1934–5. There they would serve as part of an international peacekeeping force while a plebiscite on the status of the territory was carried out under League of Nations auspices in January 1935. The context of this article is the events surrounding the creation of the peacekeeping force in December 1934.That the Irish Free State should be mentioned as a possible contributor to the international force for the Saar is an illustration of the emerging mediatory role the state was to adopt after its three-year term on the League Council concluded in September 1933. With an Irish diplomat, Sean Lester, seconded to League service as High Commissioner in Danzig from 1934, and with Irish-born Edward Phelan, Assistant Director of the International Labour Organisation, being mentioned as a possible contender for the League post of Deputy Secretary-General in 1933, and with Eamon de Valera rising in importance as an international statesman and League supporter, Ireland’s involvement in the Saar was both an illustration and a result of the state’s prominent position in the League in the early to mid-1930s.

Scribes, Cooks, and Artisans: Breaking Zhou Tradition

Early China ◽  
1995 ◽  
Vol 20 ◽  
pp. 241-277 ◽  
Author(s):  
Constance A. Cook
Keyword(s):  
The State ◽  
The Other ◽  
Political Control ◽  
Other Hand ◽  
Bronze Vessels ◽  
Western Zhou ◽  
The Creation ◽  
Bronze Inscriptions ◽  
Eastern Zhou

Bronze Inscriptions of the Western Zhou period show how ritualists were once dedicated to maintaining the ritual apparatus supporting the divine authority of the royal Zhou lineage. Bronze and bamboo texts of the Eastern Zhou period reveal, on the other hand, that ritualists able to manipulate local rulers reliant on their knowledge subsequently subverted power into their own hands. Ritualists such as scribes, cooks, and artisans were involved in the transmission of Zhou “power” through the creation and use of inscribed bronze vessels during feasts. The expansion and bureaucratization of their roles in the Chu state provided economic and ultimately political control of the state. This was particularly the case as the Chu, like the Zhou before them, fled east to escape western invaders.

A Provincial Elite in Early Modern Tuscany: Family and Power in the Creation of the State.

1998 ◽  
Vol 29 (1) ◽  
pp. 187
Author(s):  
Thomas Kuehn ◽  
Giovanna Benadusi
Keyword(s):  
Early Modern ◽  
The State ◽  
The Creation

Le armi dotte e la Rivoluzione francese: riformismo, elitismo e meritocrazia

2012 ◽  
pp. 41-63
Author(s):  
Lorenzo Cuccoli
Keyword(s):  
Eighteenth Century ◽  
Public Services ◽  
The State ◽  
Constituent Assembly ◽  
State Corporation ◽  
The Creation ◽  
École Polytechnique ◽  
The Military ◽  
The Revolution

The article focuses on the evolution of the military technical corps in France between the mid-Eighteenth century and the Restoration, and proposes for them the notion of "State corporation". This phase - an intermediate one between the corps de métier and the corps d'État - was attained first by the engineers and the artillery. These corps selected their officers by competitive examination, which functioned both as an intellectual filter and a social one. The distinction generated by this filter - nurtured by an elitist approach based on meritocracy was not overridden by the Revolution. On the contrary, it was further consecrated by the creation of the École polytechnique, which soon became controlled by the military technical corps. The "State corporation" model was then extended through the École polytechnique to the geographical engineers and the civil public services. The institutional conflicts among the technical corps during the National Constituent Assembly and those between them and the École polytechnique (1794-1799) are analyzed along these interpretative lines. While the former show their corporative resistance of geographical engineers in the name of equality, the latter bring out their corporative resistance to external education of candidates.

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