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explain terminological confusion
 
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{{for|the more general Hall algebra of a category|Ringel–Hall algebra}}
In [[mathematics]], the '''Hall algebra''' is an [[associative algebra]] with a basis corresponding to isomorphism classes of finite abelian ''p''-groups. It was first discussed by E. {{harvtxt|Steinitz|1901}} but forgotten until it was rediscovered by {{harvs|txt|authorlink=Philip Hall|first= Philip|last= Hall|year=1959}}, both of whom published no more than brief summaries of their work. The '''Hall polynomials''' are the [[structure constants]] of  the '''Hall algebra'''. The Hall algebra plays an important role in the theory of [[Masaki Kashiwara|Kashiwara]]–[[George Lusztig|Lusztig]]'s [[crystal basis|canonical bases]] in [[quantum group]]s. {{harvtxt|Ringel|1990}} generalized Hall algebras to more general categories, such as the category of representations of a [[quiver (mathematics)|quiver]].
 
== Construction ==
A [[finite set|finite]] [[abelian group|abelian]] [[p-group|''p''-group]] ''M'' is a direct sum of [[cyclic group|cyclic]] ''p''-power components <math>C_{p^{\lambda_i}},</math> where
<math>\lambda=(\lambda_1,\lambda_2,\ldots)</math> is a [[Partition (number theory)|partition]] of <math>n</math> called the ''type'' of ''M''.  Let <math>g^\lambda_{\mu,\nu}(p)</math> be the number of subgroups ''N'' of ''M'' such that ''N'' has type <math>\nu</math> and the quotient ''M/N'' has type <math>\mu</math>.  Hall proved that the functions ''g'' are [[polynomial]] functions of ''p'' with integer coefficients. Thus we may replace ''p'' with an indeterminate ''q'', which results in the '''Hall polynomials'''
 
: <math>g^\lambda_{\mu,\nu}(q)\in\mathbb{Z}[q]. \, </math>
 
Hall next constructs an [[associative ring]] <math>H</math> over <math>\mathbb{Z}[q]</math>, now called the '''Hall algebra'''. This ring has a basis consisting of the symbols <math>u_\lambda</math> and the structure constants of the multiplication in this basis are given by the Hall polynomials:
 
:<math> u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu}(q) u_\lambda. \, </math>
 
It turns out that ''H'' is a commutative ring, freely generated by the elements <math>u_{\mathbf1^n}</math> corresponding to the [[elementary abelian group|elementary ''p''-groups]]. The linear map from ''H'' to the algebra of [[symmetric function]]s defined on the generators by the formula
 
: <math>u_{\mathbf 1^n} \mapsto q^{-n(n-1)}e_n \, </math>
 
(where ''e''<sub>''n''</sub> is the ''n''th [[elementary symmetric function]]) uniquely extends to a [[ring homomorphism]] and the images of the basis elements <math>u_\lambda</math> may be interpreted via the [[Hall–Littlewood polynomial|Hall–Littlewood symmetric functions]]. Specializing ''q'' to 1, these symmetric functions become [[Schur function]]s, which are thus closely connected with the theory of Hall polynomials.
 
==References==
*{{citation|first=Philip|last=Hall|authorlink=Philip Hall|
year=1959|chapter=The algebra of partitions|title=Proceedings of the 4th Canadian mathematical congress, Banff|pages=147–159}}
* [[George Lusztig]], ''Quivers, perverse sheaves, and quantized enveloping algebras.'' J. Amer. Math. Soc. 4 (1991), no. 2, 365&ndash;421.
* {{Citation | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Symmetric functions and Hall polynomials | url=http://www.oup.com/uk/catalogue/?ci=9780198504504 | publisher=The Clarendon Press Oxford University Press | edition=2nd | series=Oxford Mathematical Monographs | isbn=978-0-19-853489-1 | mr=1354144 | year=1995}}
* {{Citation | last1=Ringel | first1=Claus Michael | title=Hall algebras and quantum groups | doi=10.1007/BF01231516 | mr=1062796 | year=1990 | journal=[[Inventiones Mathematicae]]  | volume=101 | issue=3 | pages=583–591}}
*{{citation|arxiv=math/0611617|title=Lectures on Hall algebras|last=Schiffmann|first=O|year=2006}}
*{{citation|authorlink=Ernst Steinitz|last=Steinitz|first=E.|title=Zur Theorie der Abel'schen Gruppen|journal=[[Jahresbericht der Deutschen Mathematiker-Vereinigung]]|year=1901|volume=9|pages=80–85}}
 
[[Category:Algebras]]
[[Category:Invariant theory]]
[[Category:Symmetric functions]]

Latest revision as of 03:57, 13 August 2013

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first discussed by E. Template:Harvtxt but forgotten until it was rediscovered by Template:Harvs, both of whom published no more than brief summaries of their work. The Hall polynomials are the structure constants of the Hall algebra. The Hall algebra plays an important role in the theory of KashiwaraLusztig's canonical bases in quantum groups. Template:Harvtxt generalized Hall algebras to more general categories, such as the category of representations of a quiver.

Construction

A finite abelian p-group M is a direct sum of cyclic p-power components Cpλi, where λ=(λ1,λ2,) is a partition of n called the type of M. Let gμ,νλ(p) be the number of subgroups N of M such that N has type ν and the quotient M/N has type μ. Hall proved that the functions g are polynomial functions of p with integer coefficients. Thus we may replace p with an indeterminate q, which results in the Hall polynomials

gμ,νλ(q)[q].

Hall next constructs an associative ring H over [q], now called the Hall algebra. This ring has a basis consisting of the symbols uλ and the structure constants of the multiplication in this basis are given by the Hall polynomials:

uμuν=λgμ,νλ(q)uλ.

It turns out that H is a commutative ring, freely generated by the elements u1n corresponding to the elementary p-groups. The linear map from H to the algebra of symmetric functions defined on the generators by the formula

u1nqn(n1)en

(where en is the nth elementary symmetric function) uniquely extends to a ring homomorphism and the images of the basis elements uλ may be interpreted via the Hall–Littlewood symmetric functions. Specializing q to 1, these symmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials.

References

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    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • George Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc. 4 (1991), no. 2, 365–421.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010