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Weyl character and dimension formulas for sl2

Example

For the finite-dimensional irreducible ${s l}_{2}$ -module $V (n)$ with basis $v_{0}, \dots, v_{n}$ of All irreducible finite-dimensional sl2 modules, put $χ_{n} (z) = \sum_{k = 0}^{n} z^{n - 2 k} = z^{n} + z^{n - 2} + \dots + z^{- n} (z \in C^{\times}) .$ Then for $z \neq \pm 1$ $χ_{n} (z) = \frac{z^{n + 1} - z^{- (n + 1)}}{z - z^{- 1}},$ and $\dim V (n) = n + 1.$

Facts & Assumptions

Given: The module $V (n)$ with its basis and $h$ -eigenvalues $n - 2 k$ (All irreducible finite-dimensional sl2 modules), the standard diagonal subalgebra $C h$ from The special linear Lie algebra sl_2, and the variable $z \in C^{\times}$ . In this example we define the rank-one formal character by assigning the monomial $z^{m}$ to the $h$ -eigenspace of eigenvalue $m$ and summing with eigenspace multiplicities; this convention is not attributed to the weight-space definition.

[L1]

The $h$ -eigenvalues on $V (n)$ are $n, n - 2, \dots, - n$ , each with multiplicity one, and $\dim V (n) = n + 1$ (All irreducible finite-dimensional sl2 modules, Finite-dimensional representations of sl_2, The special linear Lie algebra sl_2).

Verification

technique · direct

1.1

By [L1] the sum $χ_{n} (z) = \sum_{k = 0}^{n} z^{n - 2 k}$ is the sum of $z^{m}$ over the $h$ -eigenvalues $m$ of $V (n)$ , each counted with its multiplicity, so it is the rank-one formal character under the convention fixed in the given data.

L1given

1.2

The telescoping identity $(z - z^{- 1}) χ_{n} (z) = \sum_{k = 0}^{n} (z^{n - 2 k + 1} - z^{n - 2 k - 1}) = z^{n + 1} - z^{- (n + 1)}$ holds as an identity of Laurent polynomials.

given

2.1

For $z \neq 0$ with $z - z^{- 1} \neq 0$ , that is for $z \neq \pm 1$ , division gives $χ_{n} (z) = \frac{z^{n + 1} - z^{- (n + 1)}}{z - z^{- 1}}$ , which is the displayed formula on the regular set.

step 1.2

2.2

The identity of step 1.2 is the algebraic cancellation $z^{n + 1} - z^{- (n + 1)} = (z - z^{- 1}) χ_{n} (z)$ in the Laurent polynomial ring; it exhibits $χ_{n}$ as the quotient after cancelling the common factor $z - z^{- 1}$ , and evaluating that Laurent polynomial at $z = 1$ gives $χ_{n} (1) = n + 1$ , matching $\dim V (n) = n + 1$ by [L1].

L1step 1.2

3.1

Hence the character identity on the regular set and the dimension formula both hold, as asserted.

step 2.1step 2.2∎

Depends on

Used by

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Sources

Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)