Gennadi Sardanashvily

The three semistandard Young tableaux of shape λ = (3, 2) and weight μ = (1, 1, 2, 1). They are counted by the Kostka number K_λμ = 3.

In mathematics, the Kostka number K_λμ (depending on two integer partitions λ and μ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ and weight μ. They were introduced by the mathematician Carl Kostka in his study of symmetric functions (Template:Harvtxt).

For example, if λ = (3, 2) and μ = (1, 1, 2, 1), the Kostka number K_λμ counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows. The three such tableaux are shown at right, and K_{(3, 2) (1, 1, 2, 1)} = 3.

Examples and special cases

For any partition λ, the Kostka number K_λλ is equal to 1: the unique way to fill the Young diagram of shape λ = (λ₁, λ₂, ..., λ_m) with λ₁ copies of 1, λ₂ copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on. (This tableau is sometimes called the Yamanouchi tableau of shape λ.)

The Kostka number K_λμ is positive (i.e., there exist semistandard Young tableaux of shape λ and weight μ) if and only if λ and μ are both partitions of the same integer n and λ is larger than μ in dominance order.

In general, there are no nice formulas known for the Kostka numbers. However, some special cases are known. For example, if μ = (1, 1, 1, ..., 1) is the partition whose parts are all 1 then a semistandard Young tableau of weight μ is a standard Young tableau; the number of standard Young tableaux of a given shape λ is given by the hook-length formula.

Kostka numbers, symmetric functions and representation theory

In addition to the purely combinatorial definition above, they can also be defined as the coefficients that arise when one expresses the Schur polynomial s_λ as a linear combination of monomial symmetric functions m_μ:

$s_{\lambda }=\sum _{\mu }K_{\lambda \mu }m_{\mu }.\$

Because of the connections between symmetric function theory and representation theory, Kostka numbers also express the decomposition of the permutation module M_μ in terms of the representations V_λ corresponding to the character s_λ, i.e.,

$M_{\mu }=\bigoplus _{\lambda }K_{\lambda \mu }V_{\lambda }.$

On the level of representations of the general linear group $\mathrm {GL} _{n}(\mathbb {C} )$, the Kostka number K_λμ counts the dimension of the weight space corresponding to μ in the irreducible representation V_λ (where we require μ and λ to have at most n parts).

Examples

The Kostka numbers for partitions of size at most 3 are as follows:

K_{(0) (0)} = 1 (here (0) represents the empty partition)

K_{(1) (1)} = 1

K_{(2) (2)} = K_{(2) (1,1)} = K_{(1,1) (1,1)} = 1, K_{(1,1) (2)} = 0.

K_{(3) (3)} = K_{(3) (2,1)} = K_{(3) (2,1)} = 1

K_{(2,1) (3)} = 0, K_{(2,1) (2,1)} = 1, K_{(2,1) (1,1,1)} = 2

K_{(1,1,1) (3)} = K_{(1,1,1) (2,1)} = 0, K_{(1,1,1) (1,1,1)} = 1

These values are exactly the coefficients in the expansions of Schur functions in terms of monomial symmetric functions:

s = m = 1 (indexed by the empty partition)

s₁ = m₁

s₂ = m₂ + m₁₁

s₁₁ = m₁₁

s₃ = m₃ + m₂₁ + m₁₁₁

s₂₁ = m₂₁ + 2m₁₁₁

s₁₁₁ = m_111.

Template:Harvtxt gave tables of these numbers for partitions of numbers up to 8.

Generalizations

Kostka numbers are special values of the 1 or 2 variable Kostka polynomials:

$K_{\lambda \mu }=K_{\lambda \mu }(1)=K_{\lambda \mu }(0,1).$

Notes

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References

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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
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