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Exterior powers and fundamental weights of sl_n

Example

Assume the Axiom of Choice. Let $g = {s l}_{n} (C)$ with its diagonal Cartan $h$ , coordinate functionals $ε_{i}$ and upper-triangular positive system as in Standard and dual representations of sl_n. For every $k$ with $1 \leq k \leq n - 1$ the exterior power $Λ^{k} (C^{n})$ is an irreducible module of highest weight $ω_{k} = ε_{1} + \dots + ε_{k}$ (Fundamental weights).

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ , the standard module $V = C^{n}$ with basis $e_{1}, \dots, e_{n}$ , and $W = Λ^{k} (V)$ with basis the wedges $e_{I} = e_{i_{1}} \land \dots \land e_{i_{k}}$ for increasing index sets $I = {i_{1} < \dots < i_{k}} \subseteq {1, \dots, n}$ .

[A1]

The Axiom of Choice is assumed; it enters through the root-space and highest-weight theory used below (The Axiom of Choice).

[L1]

The weights of the standard module are $ε_{1}, \dots, ε_{n}$ , $ε_{1} = ω_{1}$ , and $ε_{1} + \dots + ε_{k} = ω_{k}$ for $1 \leq k \leq n - 1$ , because the simple coroots are $h_{α_{j}} = E_{j j} - E_{j + 1, j + 1}$ and $(ε_{1} + \dots + ε_{k}) (h_{α_{j}}) = δ_{k j}$ (Standard and dual representations of sl_n, Fundamental weights).

[L2]

The wedge $e_{I}$ is a weight vector of weight $\sum_{i \in I} ε_{i}$ , these weights are pairwise distinct for distinct index sets $I$ , and $E_{a b} \cdot e_{I} = \sum_{l : i_{l} = b} e_{i_{1}} \land \dots \land e_{a} \land \dots \land e_{i_{k}}$ , the sum being zero when the replacement produces a repeated index (Weight and weight space, Root systems of the classical complex Lie algebras).

[L3]

For a finite set of pairwise distinct weights and one of them, an element of $U (h)$ acts as the projection onto the corresponding weight component, since $U (h)$ is the polynomial algebra on $h^{*}$ (Poincaré–Birkhoff–Witt theorem).

[L4]

A nonzero submodule of an irreducible module is the whole module (Irreducible, completely reducible, and faithful representations, Highest-weight vectors and modules).

Verification

technique · direct

1.1

The vector $e_{{1, \dots, k}} = e_{1} \land \dots \land e_{k}$ has weight $ε_{1} + \dots + ε_{k} = ω_{k}$ by [L1] and [L2], and it is killed by every positive root vector $E_{i j}$ with $i < j$ : if $j \leq k$ then $i < j \leq k$ gives $i \in {1, \dots, k}$ and the replacement repeats an index, while if $j > k$ then $e_{j}$ is not a factor at all; either way $E_{i j} \cdot e_{{1, \dots, k}} = 0$ by [L2].

L1L2A1

2.1

From any basis wedge $e_{I} \neq e_{{1, \dots, k}}$ one reaches $e_{{1, \dots, k}}$ by positive root vectors: let $j$ be the smallest positive integer not in $I$ , so $j \leq k$ , and choose $i \in I$ with $i > j$ , which exists since $I$ has $k$ elements; then $E_{j i} \cdot e_{I} = \pm e_{I^{'}}$ with $I^{'} = I ∖ {i} \cup {j}$ a nonvanishing basis wedge whose index sum is strictly smaller, and repeating finitely many times reaches ${1, \dots, k}$ .

L2step 1.1

2.2

From $e_{{1, \dots, k}}$ one reaches every basis wedge by negative root vectors: for $a \in {1, \dots, k}$ and $b > k$ the operator $E_{b a}$ replaces the factor $e_{a}$ by $e_{b}$ without repeated indices (as $b \notin {1, \dots, k}$ ), giving $\pm e_{I^{''}}$ with $I^{''} = I_{0} ∖ {a} \cup {b}$ ; successive replacements of this kind produce every increasing index set.

L2step 1.1

3.1

$W$ is irreducible: if $0 \neq U \subseteq W$ is a submodule, then by [L3] some basis wedge $e_{I}$ lies in $U$ , so by step 2.1 the highest vector $e_{{1, \dots, k}}$ lies in $U$ , and by step 2.2 every basis wedge lies in $U$ ; hence $U = W$ by [L4].

L3L4step 2.1step 2.2

4.1

By step 1.1 the vector $e_{{1, \dots, k}}$ is a highest weight vector of weight $ω_{k}$ , and by step 3.1 the module is irreducible; hence $Λ^{k} (C^{n})$ has highest weight $ω_{k}$ , as asserted.

step 1.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

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