In representation theory, a branch of mathematics, the Kostant partition function, introduced by Bertram Kostant (1958, 1959), of a root system is the number of ways one can represent a vector (weight) as a non-negative integer linear combination of the positive roots . Kostant used it to rewrite the Weyl character formula as a formula (the Kostant multiplicity formula) for the multiplicity of a weight of an irreducible representation of a semisimple Lie algebra. An alternative formula, that is more computationally efficient in some cases, is Freudenthal's formula.

The Kostant partition function can also be defined for Kac–Moody algebras and has similar properties.

Examples

A2

The Kostant partition function for the A2 root system
Values of the Kostant partition function on the span of the positive roots for the root system B_2
Values of the Kostant partition function for the root system . The root system is given the Euclidean coordinates .

Consider the A2 root system, with positive roots , , and . If an element can be expressed as a non-negative integer linear combination of , , and , then since , it can also be expressed as a non-negative integer linear combination of the positive simple roots and :

with and being non-negative integers. This expression gives one way to write as a non-negative integer combination of positive roots; other expressions can be obtained by replacing with some number of times. We can do the replacement times, where . Thus, if the Kostant partition function is denoted by , we obtain the formula

.

This result is shown graphically in the image at right. If an element is not of the form , then .

B2

The partition function for the other rank 2 root systems are more complicated but are known explicitly.[1][2]

For B2, the positive simple roots are , and the positive roots are the simple roots together with and . The partition function can be viewed as a function of two non-negative integers and , which represent the element . Then the partition function can be defined piecewise with the help of two auxiliary functions.

If , then . If , then . If , then . The auxiliary functions are defined for and are given by and for even, for odd.

G2

For G2, the positive roots are and , with denoting the short simple root and denoting the long simple root.

The partition function is defined piecewise with the domain divided into five regions, with the help of two auxiliary functions.

Relation to the Weyl character formula

Inverting the Weyl denominator

For each root and each , we can formally apply the formula for the sum of a geometric series to obtain

where we do not worry about convergence—that is, the equality is understood at the level of formal power series. Using Weyl's denominator formula

we obtain a formal expression for the reciprocal of the Weyl denominator:[3]

Here, the first equality is by taking a product over the positive roots of the geometric series formula and the second equality is by counting all the ways a given exponential can occur in the product. The function is zero if the argument is a rotation and one if the argument is a reflection.

Rewriting the character formula

This argument shows that we can convert the Weyl character formula for the irreducible representation with highest weight :

from a quotient to a product:

The multiplicity formula

Using the preceding rewriting of the character formula, it is relatively easy to write the character as a sum of exponentials. The coefficients of these exponentials are the multiplicities of the corresponding weights. We thus obtain a formula for the multiplicity of a given weight in the irreducible representation with highest weight :[4]

.

This result is the Kostant multiplicity formula.

The dominant term in this formula is the term ; the contribution of this term is , which is just the multiplicity of in the Verma module with highest weight . If is sufficiently far inside the fundamental Weyl chamber and is sufficiently close to , it may happen that all other terms in the formula are zero. Specifically, unless is higher than , the value of the Kostant partition function on will be zero. Thus, although the sum is nominally over the whole Weyl group, in most cases, the number of nonzero terms is smaller than the order of the Weyl group.

References

  1. Tarski, Jan; University of California, Berkeley. (April 1963). "Partition Function for Certain Simple Lie Algebras". Journal of Mathematical Physics. United States Air Force, Office of Scientific Research. 4 (4): 569–574. doi:10.1063/1.1703992. hdl:2027/mdp.39015095253541. Retrieved 4 June 2023.
  2. Capparelli, Stefano (2003). "Calcolo della funzione di partizione di Kostant". Bollettino dell'Unione Matematica Italiana. 6-B (1): 89–110. ISSN 0392-4041.
  3. Hall 2015 Proposition 10.27
  4. Hall 2015 Theorem 10.29

Sources

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