# Tagged Questions

**2**

votes

**1**answer

209 views

### Truncated induction for exceptional cases

In Carter's book (Finite groups of Lie type), he reviews the truncated induction procedure (called j-operation in the text) of Macdonald-Lusztig-Spaltenstein in great detail for the classical Weyl ...

**1**

vote

**1**answer

122 views

### The orbit $(G\cdot X) \cap \mathfrak{t}$ for $X\in \mathfrak{t}$ singular

This question may be a simple problem for experts. Let $G$ be a connected compact Lie group and $T$ be its maximal torus. Let $\mathfrak{g}$ and $\mathfrak{t}$ be the corresponding Lie algebras. We ...

**3**

votes

**1**answer

190 views

### 'Generalised' coinvariant algebras

Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, ...

**2**

votes

**2**answers

240 views

### For a Weyl group, what is the connection between its exponents and lengths of its elements?

The following seems to be true: if $|W_q| := \sum {q^{l(w)}}$, where the sum is taken over the elements $w$, then $|W_q| = \prod {(1 + q +...+ q^{e_i})}$, where the product is taken over the exponents ...

**10**

votes

**5**answers

644 views

### About the intrinsic definition of the Weyl group of complex semisimple Lie algebras

It may be a easy question for experts.
The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra ...

**1**

vote

**0**answers

177 views

### Why Weyl group associated to Cartan matrix which defines positive definite bilinear form is finite?

In the book by V.Kac "Infinite dimensional Lie algebras" in proposition 4.9 it is argued that Weyl group constructed by Cartan matrix which defines positive-definite metric on Cartan subalgebra is ...

**6**

votes

**3**answers

386 views

### Occurrences of a simple reflection in the longest element of a Weyl group?

While looking at a preprint I've just bumped into a question about the longest element $w_0$ of a Weyl group $W$ (say irreducible of a Lie type $A$ - $G$ and of rank $n>1$, to simplify). ...

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votes

**0**answers

457 views

### Combinatorial identity involving the Coxeter numbers of root systems

The setup is:
$R$ = irreducible (reduced) root system;
$D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$;
$\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$;
...

**9**

votes

**6**answers

1k views

### Polynomial invariants of the exceptional Weyl groups

Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let ...

**9**

votes

**4**answers

676 views

### Longest Element of an Affine Weyl Group

I know that the Weyl groups of affine Lie algebras don't have a longest element, but are there any good substitutes for w_0. In particular, is there any good substitute for a reduced decomposition of ...