ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 03 Jan 2022 17:34:56 +0100Dimension of weight spaces of Lie algebra representationhttps://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/Consider a Lie algebra $\frak{g}$. Let $\lambda$ be a dominant integral weight and $L(\lambda)$ be the unique irreducible representation of highest weight $\lambda$. (Since $\lambda$ is dominant and integral, $L(\lambda)$ is finite dimensional).
We know that $L(\lambda)$ decomposes into a direct sum
$$L(\lambda)=\bigoplus_{\mu} L(\lambda)_\mu$$
where $L(\lambda)_\mu$ is a weight space of weight $\mu$.
Is there a way to compute $\dim L(\lambda)_\mu$ in Sage?
I know that Freudenthal formula can be used to find these dimensions by hand. But I want to verify if my calculations are correct. Thanks in advance!
Mon, 03 Jan 2022 14:07:51 +0100https://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/Answer by FrédéricC for <p>Consider a Lie algebra $\frak{g}$. Let $\lambda$ be a dominant integral weight and $L(\lambda)$ be the unique irreducible representation of highest weight $\lambda$. (Since $\lambda$ is dominant and integral, $L(\lambda)$ is finite dimensional).</p>
<p>We know that $L(\lambda)$ decomposes into a direct sum
$$L(\lambda)=\bigoplus_{\mu} L(\lambda)_\mu$$
where $L(\lambda)_\mu$ is a weight space of weight $\mu$. </p>
<p>Is there a way to compute $\dim L(\lambda)_\mu$ in Sage?</p>
<p>I know that Freudenthal formula can be used to find these dimensions by hand. But I want to verify if my calculations are correct. Thanks in advance!</p>
https://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/?answer=60531#post-id-60531Like this
sage: W = WeylCharacterRing(['A', 1])
sage: L4 = W([4])
sage: L4.weight_multiplicities()
{(4, 0): 1, (3, 1): 1, (2, 2): 1, (1, 3): 1, (0, 4): 1}
Mon, 03 Jan 2022 14:22:48 +0100https://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/?answer=60531#post-id-60531Comment by FrédéricC for <p>Like this</p>
<pre><code>sage: W = WeylCharacterRing(['A', 1])
sage: L4 = W([4])
sage: L4.weight_multiplicities()
{(4, 0): 1, (3, 1): 1, (2, 2): 1, (1, 3): 1, (0, 4): 1}
</code></pre>
https://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/?comment=60536#post-id-60536excerpt: "For type A (also G2, F4, E6 and E7) we will take as the weight lattice not the weight lattice of the semisimple group, but for a larger one. For type A, this means we are concerned with the representation theory of K=U(n) or G=GL(n,C) rather than SU(n) or SU(n,C)."Mon, 03 Jan 2022 17:34:56 +0100https://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/?comment=60536#post-id-60536Comment by FrédéricC for <p>Like this</p>
<pre><code>sage: W = WeylCharacterRing(['A', 1])
sage: L4 = W([4])
sage: L4.weight_multiplicities()
{(4, 0): 1, (3, 1): 1, (2, 2): 1, (1, 3): 1, (0, 4): 1}
</code></pre>
https://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/?comment=60535#post-id-60535documentation is here : https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/weyl_characters.htmlMon, 03 Jan 2022 17:32:10 +0100https://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/?comment=60535#post-id-60535Comment by slartibartfast for <p>Like this</p>
<pre><code>sage: W = WeylCharacterRing(['A', 1])
sage: L4 = W([4])
sage: L4.weight_multiplicities()
{(4, 0): 1, (3, 1): 1, (2, 2): 1, (1, 3): 1, (0, 4): 1}
</code></pre>
https://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/?comment=60533#post-id-60533Thanks a lot! I am afraid I have some further questions. My understanding is that for $\frak{sl}_2$, $\alpha_1=2\varpi_1$, where $\varpi_1$ is the unique fundamental weight and $\alpha_1$ is the unique positive root. But according to sage, the output of `W.simple_roots()` is `Finite family {1: (1, -1)}` and the output of `W.fundamental_weights()` is `Finite family {1: (1, 0)}`. But $(1,-1)\neq 2 \cdot (1,0)$. Am I missing something?Mon, 03 Jan 2022 15:23:18 +0100https://ask.sagemath.org/question/60530/dimension-of-weight-spaces-of-lie-algebra-representation/?comment=60533#post-id-60533