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Results tagged with lie-algebras
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user 25028
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
13
votes
About the definition of E8, and Rosenfeld's "Geometry of Lie groups"
Recently Lusztig gave a much simpler definition of $E_8$ (and all the simple Lie algebras/groups) that avoids the usual sign issue with the standard Chevalley/Serre construction. See Lusztig - On conj …
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3
votes
For a fixed dominant weight $\lambda$, are almost all dominant weights in the same coset abo...
Here's basically the same answer as Mikko Korhonen, but written in a way that's slightly easier for me to understand.
Let me use $\leq$ to denote the partial order on all of the vector space $E$ with …
2
votes
Accepted
Confusion about $\lambda\in\mathfrak{h}^*$ such that $L(\lambda)\in\mathcal{O}^\mathfrak{p}$
Just think about the dual parabolic root system $\Phi^\vee_I := \{\alpha^\vee\colon \alpha \in \Phi_I\}$. You can choose $\{\alpha^\vee\colon \alpha\in I\}$ to be a set of simple roots for $\Phi^\vee_ …
- 24.2k
5
votes
Tensoring $\frak{g}$-modules by fundamental representations
Joel Kamnitzer explained that $\omega_k$ is such a fundamental representation iff $0$ is a weight of the corresponding representation. In this answer I want to explain that such a weight exists for an …
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13
votes
Accepted
Dimensions of $\frak{sl}_n$-representations
Recall that the dimension of the $\mathfrak{sl}_n$ representation indexed by partition $\lambda$ is the number of semistandard Young tableaux of shape $\lambda$ with entries in $\{1,2,\ldots,n\}$.
You …
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14
votes
Accepted
Can the numerator in Weyl's character formula be written as a determinant?
The classical definition of the Schur polynomials, which considerably predates the Weyl character formula, is as a ratio of two determinants (a so-called "bialternant"): see, e.g., https://en.wikipedi …
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2
votes
Accepted
Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules
I'm not quite sure this question rises to the level of MathOverflow, which is why I initially posted only a comment, but at the request of the question-asker I am converting my comment to an answer.
F …
- 24.2k
9
votes
Why aren't $B_n$ and $C_n$ the other way around?
"Historical convention" (going back to Lie?) is probably the correct explanation, but note that under what I would call the "standard combinatorial folding procedure" as described by Stembridge in Fol …
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4
votes
2
answers
346
views
Confusion over spin representation and coordinate ring of orthogonal Grassmannian
This is a copy from MSE where the question did not attract much attention.
I'm working over $\mathbb{C}$ here. Let $G=\mathrm{SO}(2n+1)$ be the odd orthogonal group, and $P$ be the maximal parabolic …
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6
votes
Accepted
Difference of adjacent dominant weights is a root?
This is true. See "The partial order of dominant weights" by John Stembridge, 1998 (https://www.sciencedirect.com/science/article/pii/S0001870898917364). In particular look at Corollary 2.7. Alternati …
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4
votes
Accepted
Weyl Group Element $w$ fixing a root, and its presentation as product of simple reflections ...
This is definitely not true. For instance, already in $\Phi=B_2$, each root has a root orthogonal to it, so for every root there is some nontrivial element (in fact, a reflection) of the Weyl group fi …
- 24.2k
0
votes
Duals of the spinor representations of $\frak{so}_{2n}$
I don't think there is a really satisfying "conceptual" explanation of why the longest element is $-1$ times the Dynkin diagram automorphism for $D_n$ with $n$ odd and is $-1$ times the identity for $ …
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6
votes
Accepted
Motivation behind Panyushev's "constant-averages-along-orbits" conjecture
In "The root poset and its relatives" (https://arxiv.org/abs/math/0502385) Panyushev established (see Corollary 3.4) that the average size of an antichain of the root poset $\Phi^+$ of an irreducible …
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13
votes
2
answers
1k
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Significance of half-sum of positive roots belonging to root lattice?
Let $\Phi$ be a (crystallographic) root system and $\Phi^{+}$ a choice of positive roots, with $\Delta$ the corresponding choice of simple roots. So the root lattice of $\Phi$ is just $\mathbb{Z}\Delt …
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7
votes
Accepted
Existence of a weight of a representation in the fundamental Weyl chamber
Here's maybe another (more conceptual?) way to think about it.
First of all, if $\mu_1$ is a dominant weight which appears with nonzero multiplicty in $V^{\lambda_1}$, and $\mu_2$ is a dominant weigh …
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