Tagged Questions

1
vote
1answer
71 views

Decomposition into irreducibles of symmetric powers of irreps.

Suppose I have an irreducible representation of a simple Lie algebra, say $\mathfrak{sl}(n)$ or $\mathfrak{so}(n)$ i.e., $A$ and $D$ type. Given such a representation, $\Gamma_\lam …
1
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2answers
73 views

Symmetric sums and Representations of SO(3)

I had tried to help someone on math.StackExchange to prove the identity: $$ (1-Tr(A))^2+\sum_{1\le i\le j\le 3}(a_{ij}-a_{ji})^2=4$$ I guess you could argue the left hand side is …
20
votes
2answers
632 views

How much of character theory can be done without Schur’s lemma or the Artin-Wedderburn theorem?

This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question wi …
1
vote
1answer
166 views

$P^1$ minus k points

For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write $$ P^1 \setminus S \cong \mathbb{H}/G $$ where $\mathbb{H}$ is the upper-half plane …
0
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0answers
87 views

Equivariant $K$-theory, singular vectors, and flag manifolds

For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_ …
3
votes
2answers
132 views

Quotients in Sums of Rings

Suppose we are given a commutative ring R with unit-element. Now we have a composition of R as the direct product of two rings $R\cong R_1\times R_2$. It is now straight forward, …
0
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1answer
118 views

Sage or Magma Implementation of Nilpotent Orbit Varieties

For a given partition $[n_{1},...,n_{k}]$ of $N \in \mathbb{N}$ there exists a corresponding nilpotent orbit variety $O_{[n_{1},...,n_{k}]}$ in $\mathfrak{gl}(N)$ which can be repr …
0
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1answer
93 views

Free resolution for Lie algebras (reference)

What is a reference for the subject of "free resolutions for Lie algebras"? Does the term "standard resolutions" means "free resolutions"? What is a "bar resolution"? Is there o …
2
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0answers
60 views

affine schubert cells and bruhat order

Let $G$ asimply connected group over $k=\bar{k}$, $B$ a Borel subgroup and $I$ the corresponding Iwahori in G(k[[t]]), $T$ a maximal torus and $K=G(k[[t]])$. Let $\lambda\in X_{*} …
3
votes
4answers
367 views

A catalog of faithful representations of finite groups?

I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\ma …
2
votes
1answer
87 views

affine weyl group and affine schubert cells

Let $G$ a connected reductive split group over $k=\bar{k}$, $(B,T)$ a split Borel pair. Let $F:=k((t)))$. Let $\tilde{W}$ the extended Weyl group, $\tilde{W}=N_{G}(T(F))/T(O)$. B …
4
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1answer
140 views

Good effective versions of theorems of Artin and Brauer

The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups. For example, Artin's theorem is the statement that for every …
0
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0answers
53 views

How to detect if a subgroup lands inside an orthogonal group?

Equivalently, my question may be phrased as, "Are there defining characteristics of representations of orthogonal (symmetric form-preserving) groups?" Here I am working with a un …
2
votes
1answer
95 views

Reference request : dimensions of real representations of Lie groups

What is a good reference to learn about real representations of Lie groups ? I've parsed through the very enlightening book of Fulton and Harris, but it is extremely (if not exclus …
2
votes
1answer
131 views

The Jantzen-Schaper theorem

Does anybody have an electronic copy of Schaper's PhD thesis: K.D. SCHAPER, ‘Charakterformeln fur Weyl-Moduln und Specht-Moduln in Primcharacteristik’, Diplomarbeit, Bonn, 1981. …

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