# Questions tagged [weyl-group]

The Weyl group of a root system is a subgroup generated by reflections through the hyperplanes orthogonal to the roots.

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questions with no upvoted or accepted answers

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### 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 $0,1,2,...,r$;
$\...

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### Reference for class of involutions containing longest element of finite Coxeter group?

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...

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### Two-sided cells, special nilpotent orbits and special representations

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. This question concerns three classical objects of representation theory: the two-sided Kazhdan-Lusztig cells of the Weyl group $W$ of $\mathfrak{...

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### Fake degrees: why coinvariant algebra and classical groups over finite fields?

Apologies if this is not research level math (in that it concerns well-known stuff), but I am having trouble tracking down sources that explain the following. References would be very appreciated.
...

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### Action of Weyl group on regions of Shi arrangement

This is an elaboration of a question which was aked on MO several years ago, which was unanswered but deleted by the question-asker. I hope it is okay to elaborate on a deleted question like this; for ...

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### Why the hyperoctahedral group is a ``reductive'' group?

Sorry for the misleading title, I actually mean the following:
The $n$-th hyperoctahedral group, also known as the Weyl group of $\mathrm{Sp}_{2n}$ and of $\mathrm{SO}_{2n+1}$, is isomorphic to the ...

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262 views

### Name for an involution associated to a Coxeter element

Let $(W,S)$ be a finite Coxeter system, and $c \in W$ a Coxeter element.
There is an involution $g\in W$ for which the involutive map $w \mapsto gw^{-1}g$ fixes $c$. Is there a standard name for this ...

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### Two seemingly different definitions of a left cell

This is a question about two seemingly different notions of a left cell in a finite Weyl group and why they are the same. My question arose from reading a paper of W. McGovern titled "Left cells and ...

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240 views

### Counting points on Hessenberg varieties over a finite field

Let $G$ be an connected reductive group over finite field $k$. I will assume that $\text{char}(k)$ is very good for $G$ (or even larger, if preferred). Let $B\subset G$ be a Borel subgroup defined ...

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### Is there a smooth $W_{G_2}$-equivariant map from the flag manifold of $U(4)$ to that of $G_2$?

The Weyl group $W$ of $G_2$, is a group of order $12$ which is generated by the subgroup of permutations of $e_1$, $e_2$ and $e_3$ and by by the element $\tau$ which maps $(e_1,e_2,e_3)$ to $(-e_1,-...

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### Weyl Group Action on Littelmann Paths

In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via
$$
\tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi)...

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### Can the Weyl orbits of fundamental weights tell us the Cartan matrix?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\Delta$ its root system contained in $\mathfrak{t}^{\vee}$ for a Cartan sub-algebra $\mathfrak{t}$ of $\mathfrak{g}$. Let $W$ be its Weyl group....

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### Is there a geometric interpretation of the Hecke algebra of a Weyl group?

The Weyl group $W$ can be realized as a fundamental group of a topological space. So that its group algebra $\mathbb{Z}(W)$ has a geometric meaning. On the other hand, the Hecke algebra is a ...

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### Are there mathematical/physical applications of these Weyl equivariant maps?

Let $G$ be a compact Lie group and $T$ a choice of maximal torus. Denote the corresponding Lie algebras by $\mathfrak{g}$ and $\mathfrak{t}$. Elements of $\mathfrak{t} \otimes \mathbb{R}^3$ are called ...

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### Kazhdan-Lusztig basis elements appearing in product with distinguished involution

My apologies if the below is too malformed to make sense.
Let $(W,S)$ be the affine Weyl group of a reductive group $G$, and let $\{C_w\}$ be the Kazhdan-Lusztig $C$-basis (an answer in terms of the $...

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### Do these Zariski-dense subgroups of complex Chevalley group have non-empty intersection with this Bruhat cell?

Let $G$ be a complex Chevalley group (not necessarily adjoint type) with $\operatorname{\mathbb{C}-rank}\geq2$ and let $H$ be a normal subgroup of $G(\mathbb Z)$ with a finite index (so $H$ is Zariski ...

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133 views

### Is one of the hyperplane partitions of a irreducible root system always generate the whole Weyl group?

Let $\Delta$ be a irreducible root system and $\Delta^+$ be its positive roots.
We say a subset $\Delta^{\prime}\subset \Delta^+$ can generate the Weyl group if reflections of roots in $\Delta^{\...

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### Weyl theorem - possible corollary - alternative characterization of projective representation of $Z_N\times Z_N$

For an integer $N$, let $\omega=e^{2i\pi/N}$ and $A$, $B$ be the clock and shift operators:
$A=\left(\begin{matrix}
1 & 0 & \cdots & 0 \\
0 & \omega & \cdots & 0 \\
\vdots &...

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175 views

### Reference request for generalized root systems

Where can I find information on root systems where the inner product
is other than the standard (all positive) signature?

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### Weyl groups of Levi factors and their cosets

Let $G$ be a connected semisimple algebraic group defined over an algebraically closed field $F$ of characteristic $0$. Let $B$ be a minimal parabolic subgroup of $G$ (i.e. a Borel subgroup), let $P$ ...

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### Highest (short) roots and commutation relations in (twisted) DAHA

I am trying to understand, explicitly, the commutation relation between $X_\vartheta$ and $Y_\vartheta = T_0T_{s_\vartheta}$ in the (twisted) DAHA for a root system $R$, where $\vartheta$ is the ...

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### Topological obstruction to icosahedral symmetry?

Let $G$ be a compact simple lie group of rank $n$. Then the Poincaré series of $G$ is given by $$P(G,q)=\prod_{i=1}^n (1+q^{2d_i-1}),$$
where the integers $d_1\leq d_2\leq \cdots \leq d_n$ are the ...

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### Weyl group stabilizer of semisimple element in adjoint group

Let $G$ be semisimple group over $\mathbb{C}$ of adjoint type. Let $T$ be a maximal torus, $s\in T$ semisimple element. Let $W$ be a Weyl group and $W(s)$ be a stabilizer of $s$ in $W$. I am ...

**3**

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### Comparison of length functions on Weyl groups

Let $G$ be a connected reductive group over an algebraically closed field $k$ (with nice enough characteristic), and let $\sigma:G\to G$ be a finite order automorphism of $G$. The connected component $...

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### Relationship bewteen Kazhdan-Lusztig Vogan polynomial and Kazhdan-Lusztig polynomial

Let $W^I=\{w\in W: w^{-1}\Phi_I^+\subseteq \Phi^+\}$, $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$
Let $M(\lambda)$ be the Verma module with highest weight $\...

**3**

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### Cohomology of root data

Let $(X,\Phi,X^\vee,\Phi^\vee)$ be a semisimple root datum (in the sense of SGAIII), and $W_0$ its (finite) Weyl group. What is known about the cohomology groups $H^n(W_0, X^\vee)$ ?

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### How do I obtain Vandermonde identity from Weyl's denominator formula?

Let $\Phi$ be a root system with Weyl group $\operatorname{Weyl}(\Phi)$, let $\Phi^+$ be a set of positive roots for $\Phi$ and $\rho$ be the half sum of the elements of $\Phi^+$. Then the Weyl's ...

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### Computing Springer action on the homology of affine Springer fibers

Lusztig defined (in Sec. 5, also Sage) a Springer action of the affine Weyl group on the homology of affine Springer fibers (Iwahori one, i.e. in an affine flag variety). In the regular semisimple ...

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### Cell modules for type D Weyl group

Does anyone know a reference for the construction of cell modules for the Weyl group of type $D$, without any reference to the Weyl group of type $B$?
What would be even better is an idempotent ...

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### Tangent space of a curve isomorphic to $\mathbb{P}^1$ at a $T$-fixed point

Let $G$ be a reductive group defined over a field $k$ of characteristic zero, with maximal split torus $T$, Borel $B \supset T$ defining a set of simple roots $\Delta$. Furthermore denote by $W$ the ...

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### On $\Psi$-generating paths in the Bruhat order of a Weyl group

Let $W$ be a Weyl group with roots $R$ and positive roots $R^+$. Let $v\in W$ of length $r$. We call $\mathbb{m}=(\alpha_1,\ldots,\alpha_r)\in(R^+)^r$ a Bruhat path from $1$ to $v$ if $1\lessdot s_{\...

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### Carter Payne homomorphisms and reduced expressions

Let $G$ be an algebraic group and $W$ denote the underlying affine Weyl group. I will label representations of the principal block of $G$ by their alcoves, which in turn I label by the corresponding ...

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### Deodhar's inequality: when the equality holds?

Let $(W,S)$ be a Coxeter system, $T=\bigcup_{w\in W}wSw^{-1}$ and $\ell$ be the length function.
It is well-known that one have the following Deodhar's inequality:
Let $x\le y\le w$. Then
$|\{r\...

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177 views

### The growth of maximum elements for the reflection group $D_n$

Let's consider maximum coefficients in Poincaré polynomial (or growth series) for the reflection group (or Weyl group, or finite Coxeter group) $D_n$ as mentioned in A162206.
The maximal numbers $M(n)$...

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### Denominator identity for Lie superalgebras

Let $\mathfrak g$ be a basic classic simple Lie superalgebra.
Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...

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### odd positive roots of basic classical Lie superalgebras

In the Lie algebra case, positive roots are "almost ($S_{\alpha}$ permutes $\Delta^+ -\{\alpha\}$)" invariant under simple reflections. A similar statement I want to understand for Lie superalgebras.
...

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### Combinatorial and computational problem related to Weyl groups and the coroot lattice

Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\...

**2**

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### Description of the center of a reductive group using absolute and relative roots

Let $G$ be a connected, reductive group over a field $k$. Let $T \subseteq B$ be a maximal torus and Borel subgroup of $G$ with corresponding base $\Delta \subseteq X(T)$. Then $T$ contains $Z(G)$, ...

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### The mod p reduction of Weyl groups and their polynomial invariants

Let $C$ be an $n\times n$ Cartan matrix, then the Weyl group $W\subset GL(n,Z)$.
Let $W_p\subset GL(n,Z_p)$ be the mod p reduction of $W$. I want to know whether there are known work on the structure ...

**2**

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**2**answers

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### Greatest element of ${}^IW$

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$.
Let $W$ be the associated Weyl group and let $\Phi$ be its root system.
We write $\Phi^+...

**2**

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**1**answer

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### The simple reflections of the Weyl group in $\operatorname{SO}_{2n}(\mathbb C)$

Let $W$ be the Weyl group corresponding to the maximal torus $diag(t_1, . . . , t_{n}, t^{−1}_n, . . . , t^{−1}_1)$ in a Borel group of $\operatorname{SO}_{2n}(\mathbb C)$.
What are the matrices ...

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191 views

### Condition for a sum of images of fundamental dominant weights to lie on a wall

Let $\Delta$ be a system of simple roots in a root system with Weyl group $W$. For $\alpha\in\Delta$, let $\varpi_\alpha$ be the corresponding fundamental dominant weight. Let $w\neq r$ be elements of ...

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### Uniqueness in Mare combinatorics and bounds on Gromov-Witten invariants

Let $R$ be the root system of a Weyl group $W$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. the set of positive roots $\alpha$ such that $\ell(s_\alpha)=2\operatorname{ht}\alpha-1$. Based ...

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### Equivalence of categories between the loop algebra of $sl_{n+1}$ and the affine Weyl group of $GL_\ell(C)$

In this paper here, Theorem 4.9 page 18, Charri and Pressley are claiming that there exists an equivalence of Categories between certain categories of the Lie algebra $\tilde{ \mathfrak g}=\mathfrak{...

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### Bruhat cell of a Coxeter element

If $G$ is a complex Chevalley group and $H\leq G(\mathbb Z)$ dense in $G(\mathbb C)$, can I find $g\in H$ conjugated in $G(\mathbb Z)$ to an element in the Bruhat cell $BwB$ where $w$ represent a ...

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### Nontrivial relations of the irreducible root systems

For the root system of the type $A_n$, the roots are $\alpha _{i,j}$, $1\le i\neq j\le n$, we have the nontrivial relations $(x_{i,j} (t), x_{j,k}(u)) = x_{i,k}(tu)$ if $i, j, k$ are distinct. ($x_{i,...

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### Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$

Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$?
For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...

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93 views

### Weyl Group action on the complement of the Tits Cone in a Kac-Moody algebra

Given a Kac-Moody algebra $\mathfrak h$ and its Weyl group $W$, the action of $W$ on the Tits cone $X$ is well understood. Decompose $\mathfrak h$ into $X\cup -X\cup L$. Then the action of $W$ on $-...

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### An identity in Weyl group

Let $W$ be a Weyl group generated by the simple reflections $s_i$, $i \in I$, where $I$ is the vertex set of the Dynkin diagram of $W$. For $J \subset I$, let $W_J$ be the subgroup of $W$ generated by ...

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229 views

### Regarding Weyl's integration formula - from "equality up to scalar" to "exact equality"

Teaching a course on representations of compact groups, I want to prove Weyl's integration formula. It seems to me that in a few sources, they prove the formula up to scalar, and don't even ...