Parity-check matrix

In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by Template:Harvtxt.

Template:Harvtxt used these representations to find all representations of all finite simple groups of Lie type.

Motivation

Suppose that G is a reductive group defined over a finite field, with Frobenius map F.

Macdonald conjectured that there should be a map from general position characters of F-stable maximal tori to irreducible representations of G^F (the fixed points of F). For general linear groups this was already known by the work of Template:Harvtxt. This was the main result proved by Deligne and Lusztig; they found a virtual representation for all characters of an F-stable maximal torus, which is irreducible (up to sign) when the character is in general position.

When the maximal torus is split, these representations were well known and are given by parabolic induction of characters of the torus (extend the character to a Borel subgroup, then induce it up to G). The representations of parabolic induction can be constructed using functions on a space, which can be thought of as elements of a suitable zeroth cohomology group. Deligne and Lusztig's construction is a generalization of parabolic induction to non-split tori using higher cohomology groups. (Parabolic induction can also be done with tori of G replaced by Levi subgroups of G, and there is a generalization of Deligne–Lusztig theory to this case too.)

Drinfeld proved that the discrete series representations of SL₂(F_q) can be found in the ℓ-adic cohomology groups

H¹_c(X, Q_ℓ)

of the affine curve X defined by

xy^q−yx^q = 1.

The polynomial xy^q−yx^q is a determinant used in the construction of the Dickson invariant of the general linear group, and is an invariant of the special linear group.

The construction of Deligne and Lusztig is a generalization of this fundamental example to other groups. The affine curve X is generalized to a T^F bundle over a "Deligne–Lusztig variety" where T is a maximal torus of G, and instead of using just the first cohomology group they use an alternating sum of ℓ-adic cohomology groups with compact support to construct virtual representations.

The Deligne-Lusztig construction is formally similar to Weyl's construction of the representations of a compact group from the characters of a maximal torus. The case of compact groups is easier partly because there is only one conjugacy class of maximal tori. The Borel–Weil–Bott construction of representations of algebraic groups using coherent sheaf cohomology is also similar.

For real semisimple groups there is an analogue of the construction of Deligne and Lusztig, using Zuckerman functors to construct representations.

Deligne–Lusztig varieties

The construction of Deligne-Lusztig characters uses a family of auxiliary algebraic varieties X_T called Deligne–Lusztig varieties, constructed from a reductive linear algebraic group G defined over a finite field F_q.

If B is a Borel subgroup of G and T a maximal torus of B then we write

W_T,B

for the Weyl group (normalizer mod centralizer)

N_G(T)/T

of G with respect to T, together with the simple roots corresponding to B. If B₁ is another Borel subgroup with maximal torus T₁ then there is a canonical isomorphism from T to T₁ that identifies the two Weyl groups. So we can identify all these Weyl groups, and call it 'the' Weyl group W of G. Similarly there is a canonical isomorphism between any two maximal tori with given choice of positive roots, so we can identify all these and call it 'the' maximal torus T of G.

By the Bruhat decomposition

G = BWB,

the subgroup B₁ can be written as the conjugate of B by bw for some b∈B and w∈W (identified with W_T,B) where w is uniquely determined. In this case we say that B and B₁ are in relative position w.

Suppose that w is in the Weyl group of G, and write X for the smooth projective variety of all Borel subgroups of G. The Deligne-Lusztig variety X(w) consists of all Borel subgroups B of G such that B and F(B) are in relative position w. In other words it is the inverse image of the G-homogeneous space of pairs of Borel subgroups in relative position w, under the Lang isogeny with formula

g.F(g)⁻¹.

For example, if w=1 then X(w) is 0-dimensional and its points are the rational Borel subgroups of G.

We let T(w) be the torus T, with the rational structure for which the Frobenius is wF. The G^F conjugacy classes of F-stable maximal tori of G can be identified with the F-conjugacy classes of W, where we say w∈W is F-conjugate to elements of the form vwF(v)⁻¹ for v∈W. If the group G is split, so that F acts trivially on W, this is the same as ordinary conjugacy, but in general for non-split groups G, F may act on W via a non-trivial diagram automorphism. The F-stable conjugacy classes can be identified with elements of the non-abelian galois cohomology group of torsors

H¹(F,W).

Fix a maximal torus T of G and a Borel subgroup B containing it, both invariant under the Frobenius map F, and write U for the unipotent radical of B. If we choose a representative w′ of the normalizer N(T) representing w, then we define X′(w′) to be the elements of G/U with F(u)=uw′. This is acted on freely by T(F), and the quotient is isomorphic to X(T). So for each character θ of T(w)^F we get a corresponding local system F_θ on X(w). The Deligne-Lusztig virtual representation

R^θ(w)

of G^F is defined by the alternating sum

$R^{\theta }(w)=\sum _i(-1{)}^i{H}_c^i(X(w),F_{\theta })$

of l-adic compactly supported cohomology groups of X(w) with coefficients in the l-adic local system F_θ.

If T is a maximal F-invariant torus of G contained in a Borel subgroup B such that B and FB are in relative position w then R^θ(w) is also denoted by R^θ_T⊂B, or by R^θ_T as up to isomorphism it does not depend on the choice of B.

Properties of Deligne–Lusztig characters

• The character of R^θ_T does not depend on the choice of prime l≠p, and if θ=1 its values are rational integers.

• Every irreducible character of G^F occurs in at least one character R^θ(w).

• The inner product of R^θ_T and R^θ′_T′ is equal to the number of elements of W(T,T′)^F taking θ to θ′. The set W(T,T′) is the set of elements of G taking T to T′ under conjugation, modulo the group T^F which acts on it in the obvious way (so if T=T′ it is the Weyl group). In particular the inner product is 0 if w and w′ are not F-conjugate. If θ is in general position then R^θ_T has norm 1 and is therefore an irreducible character up to sign. So this verifies Macdonald's conjecture.

• The representation R^θ_T contains the trivial representation if and only if θ=1 (in which case the trivial representation occurs exactly once).

• The representation R^θ_T has dimension

$\dim (R_T^{\theta })=\frac{\pm \left|G^F\right|}{\left|T^F\right|\left|U^F\right|}$

where U^F is a Sylow p-subgroup of G^F, of order the largest power of p dividing |G^F|.

• The restriction of the character R^θ_T to unipotent elements u does not depend on θ and is called a Green function, denoted by Q_T,G(u) (the Green function is defined to be 0 on elements that are not unipotent). The character formula gives the character of R^θ_T in terms of Green functions of subgroups as follows:

$R_T^{\theta }(x)=\frac{1}{\left|G_s^F\right|}\sum _{g\in G^F,g^{-1}sg\in T}{Q}_{gT{g}^{-1},G_s}(u)\theta (g^{-1}sg)$

where x=su is the Jordan–Chevalley decomposition of x as the product of commuting semisimple and unipotent elements s and u, and G_s is the identity component of the centralizer of s in G. In particular the character value vanishes unless the semisimple part of x is conjugate under G^F to something in the torus T.

• The Deligne-Lusztig variety is usually affine, in particular whenever the characteristic p is larger than the Coxeter number h of the Weyl group. If it is affine and the character θ is in general position (so that the Deligne-Lusztig character is irreducible up to sign) then only one of the cohomology groups Hⁱ(X(w),F_θ) is non-zero (the one with i equal to the length of w) so this cohomology group gives a model for the irreducible representation. In general it is possible for more than one cohomology group to be non-zero, for example when θ is 1.

Lusztig's classification of irreducible characters

Lusztig classified all the irreducible characters of G^F by decomposing such a character into a semisimple character and a unipotent character (of another group) and separately classifying the semisimple and unipotent characters.

The dual group

The representations of G^F are classified using conjugacy classes of the dual group of G. A reductive group over a finite field determines a root datum (with choice of Weyl chamber) together with an action of the Frobenius element on it. The dual group G^* of a reductive algebraic group G defined over a finite field is the one with dual root datum (and adjoint Frobenius action). This is similar to the Langlands dual group (or L-group), except here the dual group is defined over a finite field rather than over the complex numbers. The dual group has the same root system, except that root systems of type B and C get exchanged.

The local Langlands conjectures state (very roughly) that representations of an algebraic group over a local field should be closely related to conjugacy classes in the Langlands dual group. Lusztig's classification of representations of reductive groups over finite fields can be thought of as a verification of an analogue of this conjecture for finite fields (though Langlands never stated his conjecture for this case).

Jordan decomposition

In this section G will be a reductive group with connected center.

An irreducible character is called unipotent if it occurs in some R¹_T, and is called semisimple if its average value on regular unipotent elements is non-zero (in which case the average value is 1 or −1). If p is a good prime for G (meaning that it does not divide any of the coefficients of roots expressed as linear combinations of simple roots) then an irreducible character is semisimple if and only if its order is not divisible by p.

An arbitrary irreducible character has a "Jordan decompostion": to it one can associate a semisimple character (corresponding to some semisimple element s of the dual group), and a unipotent representation of the centralizer of s. The dimension of the irreducible character is the product of the dimensions of its semisimple and unipotent components.

This (more or less) reduces the classification of irreducible characters to the problem of finding the semisimple and the unipotent characters.

Geometric conjugacy

Two pairs (T,θ), (T′,θ′) of a maximal torus T of G fixed by F and a character θ of T^F are called geometrically conjugate if they are conjugate under some element of G(k), where k is the algebraic closure of F_q. If an irreducible representation occurs in both R_{T^θ and RT′^θ′ then (T,θ), (T′,θ′) need not be conjugate under G^F, but are always geometrically conjugate. For example if θ = θ′ = 1 and T and T′ are not conjugate, then the identity representation occurs in both Deligne–Lusztig characters, and the corresponding pairs (T,1), (T′,1) are geometrically conjugate but not conjugate.}

The geometric conjugacy classes of pairs (T,θ) are parameterized by geometric conjugacy classes of semisimple elements s of the group G^*F of elements of the dual group G^* fixed by F. Two elements of G^*F are called geometrically conjugate if they are conjugate over the algebraic closure of the finite field; if the center of G is connected this is equivalent to conjugacy in G^*F. The number of geometric conjugacy classes of pairs (T,θ) is |Z^0F|q^l where Z⁰ is the identity component of the center Z of G and l is the semisimple rank of G.

Classification of semisimple characters

In this subsection G will be a reductive group with connected center Z. (The case when the center is not connected has some extra complications.)

The semisimple characters of G correspond to geometric conjugacy classes of pairs (T,θ) (where T is a maximal torus invariant under F and θ is a character of T^F); in fact among the irreducible characters occurring in the Deligne–Lusztig characters of a geometric conjugacy class there is exactly one semisimple character. If the center of G is connected there are |Z^F|q^l semisimple characters. If κ is a geometric conjugacy class of pairs (T,θ) then the character of the corresponding semisimple representation is given up to sign by

$\sum _{(T,\theta )\in \kappa ,{modG}^F}\frac{R_T^{\theta }}{(R_T^{\theta },R_T^{\theta })}$

and its dimension is the p′ part of the index of the centralizer of the element s of the dual group corresponding to it.

The semisimple characters are (up to sign) exactly the duals of the regular characters, under Alvis–Curtis duality, a duality operation on generalized characters. An irreducible character is called regular if it occurs in the Gelfand–Graev representation G^F, which is the representation induced from a certain "non-degenerate" 1-dimensional character of the Sylow p-subgroup. It is reducible, and any irreducible character of G^F occurs at most once in it. If κ is a geometric conjugacy class of pairs (T,θ) then the character of the corresponding regular representation is given by

$\sum _{(T,\theta )\in \kappa ,{modG}^F}\frac{{ϵ}_G{ϵ}_T{R}_T^{\theta }}{(R_T^{\theta },R_T^{\theta })}$

and its dimension is the p′ part of the index of the centralizer of the element s of the dual group corresponding to it times the p-part of the order of the centralizer.

Classification of unipotent characters

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These can be found from the cuspidal unipotent characters: those that cannot be obtained from decomposition of parabolically induced characters of smaller rank groups. The unipotent cuspidal characters were listed by Lusztig using rather complicated arguments. The number of them depends only on the type of the group and not on the underlying field; and is given as follows:

• none for groups of type A_n;
• none for groups of type ²A_n, unless n = s(s+1)/2–1 for some s, in which case there is one;
• none for groups of type B_n or C_n, unless n = s(s+1) for some s, in which case there is one (called θ₁₀ when n = 2);
• 2 for Suzuki groups of type ²B₂;
• none for groups of type D_n, unless n = s² for some even s, in which case there is one;
• none for groups of type ²D_n, unless n = s² for some odd s, in which case there is one;
• 2 for groups of type ³D₆;
• 2 for groups of type E₆;
• 3 for groups of type ²E₆;
• 2 for groups of type E₇;
• 13 for groups of type E₈;
• 7 for groups of type F₄;
• 10 for Ree groups of type ²F₄;
• 4 for groups of type G₂;
• 6 for Ree groups of type ²G₂.

The unipotent characters can be found by decomposing the characters induced from the cuspidal ones, using results of Howlett and Lehrer. The number of unipotent characters depends only on the root system of the group and not on the field (or the center). The dimension of the unipotent characters can be given by universal polynomials in the order of the ground field depending only on the root system; for example the Steinberg representation has dimension q^r, where r is the number of positive roots of the root system.

Lusztig discovered that the unipotent characters of a group G^F (with irreducible Weyl group) fall into families of size 4ⁿ (n ≥ 0), 8, 21, or 39. The characters of each family are indexed by conjugacy classes of pairs (x,σ) where x is in one of the groups Z/2Zⁿ, S₃, S₄, S₅ respectively, and σ is a representation of its centralizer. (The family of size 39 only occurs for groups of type E₈, and the family of size 21 only occurs for groups of type F₄.) The families are in turn indexed by special representations of the Weyl group, or equivalently by 2-sided cells of the Weyl group. For example, the group E₈(F_q) has 46 families of unipotent characters corresponding to the 46 special representations of the Weyl group of E₈. There are 23 families with 1 character, 18 families with 4 characters, 4 families with 8 characters, and one family with 39 characters (which includes the 13 cuspidal unipotent characters).

Examples

Suppose that q is an odd prime power, and G is the algebraic group SL₂. We describe the Deligne–Lusztig representations of the group SL₂(F_q). (The representation theory of these groups was well known long before Deligne–Lusztig theory.)

The irreducible representations are:

• The trivial representation of dimension 1.
• The Steinberg representation of dimension q
• The (q − 3)/2 irreducible principal series representations of dimension q + 1, together with 2 representations of dimension (q + 1)/2 coming from a reducible principal series representation.
• The (q − 1)/2 irreducible discrete series representations of dimension q − 1, together with 2 representations of dimension (q − 1)/2 coming from a reducible discrete series representation.

There are two classes of tori associated to the two elements (or conjugacy classes) of the Weyl group, denoted by T(1) (cyclic of order q−1) and T(w) (cyclic of order q + 1). The non-trivial element of the Weyl group acts on the characters of these tori by changing each character to its inverse. So the Weyl group fixes a character if and only if it has order 1 or 2. By the orthogonality formula, R^θ(w) is (up to sign) irreducible if θ does not have order 1 or 2, and a sum of two irreducible representations if it has order 1 or 2.

The Deligne-Lusztig variety X(1) for the split torus is 0-dimensional with q+1 points, and can be identified with the points of 1-dimensional projective space defined over F_q. The representations R^θ(1) are given as follows:

• 1+Steinberg if θ=1
• The sum of the 2 representations of dimension (q+1)/2 if θ has order 2.
• An irreducible principal series representation if θ has order greater than 2.

The Deligne-Lusztig variety X(w) for the non-split torus is 1-dimensional, and can be identified with the complenent of X(1) in 1-dimensional projective space. So it is the set of points (x:y) of projective space not fixed by the Frobenius map (x:y)→ (x^q:y^q), in other words the points with

$x{y}^q-y{x}^q\ne 0$

Drinfeld's variety of points (x,y) of affine space with

$x{y}^q-y{x}^q=1$

maps to X(w) in the obvious way, and is acted on freely by the group of q+1th roots λ of 1 (which can be identified with the elements of the non-split torus that are defined over F_q), with λ taking (x,y) to (λx,λy). The Deligne Lusztig variety is the quotient of Drinfeld's variety by this group action. The representations −R^θ(w) are given as follows:

• Steinberg−1 if θ=1
• The sum of the 2 representations of dimension (q−1)/2 if θ has order 2.
• An irreducible discrete series representation if θ has order greater than 2.

The unipotent representations are the trivial representation and the Steinberg representation, and the semisimple representations are all the representations other than the Steinberg representation. (In this case the semisimple representations do not correspond exactly to geometric conjugacy classes of the dual group, as the center of G is not connected.)

Intersection cohomology and character sheaves

Template:Harvtxt replaced the ℓ-adic cohomology used to define the Deligne-Lusztig representations with intersection ℓ-adic cohomology, and introduced certain perverse sheaves called character sheaves. The representations defined using intersection cohomology are related to those defined using ordinary cohomology by Kazhdan–Lusztig polynomials. The F-invariant irreducible character sheaves are closely related to the irreducible characters of the group G^F.

References

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  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.
  
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  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.
  
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• 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
• 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, Adv. Math. , 57 (1985) 226–265; Adv. Math. , 57 (1985)266–315; Adv. Math. , 59 (1986) 1–63; Adv. Math. , 61 (1986) 103–155
• 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