All Questions
32 questions
1
vote
1
answer
102
views
Multiplicities and double and triple tensor products of simple $\frak{g}$-modules
Given a complex simple Lie algebra $\frak{g}$ and a simple module $V_{\lambda}$ for some dominant weight $\lambda$. Consider the tensor product decomposition
$$
V_{\lambda} \otimes V_{\lambda} \simeq ...
- 417
1
vote
0
answers
108
views
Schur Weyl Duality for Maximal Torus
I wanted to know if there's some version of Schur Weyl Duality for the maximal torus $T \subset \operatorname{GL}(V)$? Is there also some combinatorial object which might be useful for the same?
2
votes
1
answer
107
views
Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces
For the special linear algebra $\frak{sl}_{m}$ which finite dimensional irreducible representations $V_{\mu}$ have non-trivial zero weight spaces?
For $\frak{sl}_2$ this is clear: $V_{2k\pi}$ for $\pi$...
- 157
3
votes
0
answers
139
views
Root space inner products and the partial order on roots
For a root system $R$ and a choice of positive roots $R^+$ it is a standard fact (see, e.g., Bourbaki, "Lie Groups and Lie Algebras," Theorem 1 of Section 1.3 of Chapter VI) that
if $(\...
3
votes
2
answers
492
views
Pairing a root with the half-sum of positive roots
Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive ...
3
votes
1
answer
280
views
Decomposition of tensor powers of the vector representation of $\frak{sl}_n$
Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as
$$
V(\pi_1) \...
- 257
1
vote
1
answer
189
views
Tensoring irreducible representations corresponding to root lattice elements
Let $\frak{g}$ be a complex semisimple Lie algebra with root lattice $Q$ and positive weight space $P^+$. Let $\lambda, \mu \in Q \cap P^+$, with corresponding respective fin-dim irreducible ...
- 327
4
votes
1
answer
239
views
Number of representations of a semisimple Lie algebra of any given dimension
For a semisimple complex Lie algebra $\frak{g}$ it is well known that irreducible finite-dimensional representation are not characterised by their dimension.
More formally, let us define an ...
- 327
4
votes
0
answers
111
views
How many diagrams interlace a given Young diagram?
For a fixed partition $\lambda=(\lambda_1\geq\dots\geq \lambda_n)$ we say $\mu=(\mu_1\geq \dots \geq \mu_{n-1})$ $\textit{interlaces}$ $\lambda$ iff
$$\lambda_1\geq \mu_1\geq \dots \geq \mu_{n-1}\geq \...
5
votes
1
answer
310
views
Non-standard partial orders on root systems
Let $\frak{g}$ be a semisimple complex Lie algebra and let $\Delta$ be its associated root system with $\{\alpha_1, \dotsc, \alpha_l\}$ a choice of positive roots. As we all know - $\Delta$ admits a ...
1
vote
0
answers
138
views
When is the zero weight space of an irreducible $\frak{sl}_{n+1}$-module non-trivial?
Take the complex semisimple Lie algebra $\frak{sl}_{n+1}$, with space of dominant integral weights $P(\frak{sl}_{n+1})$. For $V(\lambda)$ the irreducible representation corresponding to $\lambda \in P(...
- 969
2
votes
0
answers
141
views
Partial sum of Weingarten functions over symmetric group
I have a question about partial sums of Weingarten functions. The Weingarten functions are defined as
$$
E_U[U_{i_1,j_1}\dotsm U_{i_k,j_k}U^*_{i'_1,j'_1}\dotsm U^*_{i'_k,j'_k}]=\sum_{\alpha,\beta \in \...
- 121
2
votes
0
answers
135
views
Fusion rules for the Lie algebra $\frak{so}_{2n+1}$
For the Lie algebra $\mathfrak{so}_{2n+1}$ where can I find a description of the fusion rules of it fundamental representations? In more detail: For $\pi_i$ and $\pi_j$ two fundamental weights of $\...
- 393
17
votes
4
answers
2k
views
Reference request: Grassmannian and Plucker coordinates in type B, C, D
Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker ...
- 6,201
9
votes
1
answer
444
views
Young tableaux for exceptional Lie algebras
Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.
Does ...
- 835
7
votes
0
answers
166
views
"Non standard" formulas for eigenspaces in $V_\rho$
In the context of the Simple Lie Algebras Representations, let $\rho$ be half-the-sum of the positive roots and let $V_\rho$ be the irreducible representation of highest weight $\rho$.
Let$\mu$ be a ...
- 233
11
votes
0
answers
432
views
Connection between Gelfand-Tsetlin basis and SSYT basis in Schur module
Consider an $n$-dimensional complex vector space $V$ with a chosen basis $e_1,\ldots,e_n$. This basis defines a Cartan decompostion of $GL(V)\cong GL_n$ and for an (integral dominant) highest weight $\...
- 3,513
3
votes
1
answer
572
views
Dimension of the zero weight space in $V_{2\rho}$
Let $\rho$ be the half sum of the positive roots (also the sum of fundmental weights) for $SL_n(\mathbb C)$. Then $2ρ$ is in the root lattice. Then what is the dimension of the zero weight space for ...
- 31
14
votes
1
answer
503
views
Littlewood–Richardson rule and the Harish-Chandra-Itzykson-Zuber integral
The Littlewood–Richardson rule states that the product of two Schur polynomials can be written as a finite weighted sum of Schur polynomials. More precisely
$$
s_\lambda s_\mu = \sum_\nu c_{\lambda,\...
- 2,135
6
votes
2
answers
334
views
Multiplication in universal enveloping algebra in terms of PBW isomorphism
Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt ...
- 7,377
27
votes
1
answer
891
views
Why do the adjoint representations of three exceptional groups have the same first eight moments?
For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...
- 148k
7
votes
2
answers
533
views
Schur polynomial, change of variable
Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$.
Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ ...
- 3,804
3
votes
0
answers
267
views
adding a boundary to the finite upper half-plane
Let $\Bbb{F}_q$ be a finite field, let $\delta \in \Bbb{F}_q$ be a non-square, let $\Bbb{F}_{q^2} = \Bbb{F}_q\big( \sqrt{\delta} \big)$ be the corresponding quadratic extension,
and let ${\frak{H}}_q:=...
- 2,137
5
votes
0
answers
171
views
Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B
This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...
- 1,719
6
votes
1
answer
506
views
Lattice model for Affine Grassmannians of non type A
There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional $\mathbb{C}-...
- 1,719
4
votes
1
answer
907
views
Is there a generalization of Schur - Weyl duality and plethysm for direct product of special unitary groups?
Consider the semisimple compact group $K=SU(N_1)\times SU(N_2) \times \ldots \times SU(N_S)$ acting naturally on $\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \ldots \otimes \mathcal{H}_S$, ...
- 965
6
votes
2
answers
315
views
Covering relations in $K\backslash G/B$
Let $G$ be a simply connected complex Lie group, $\theta$ an involution,
and $K = G^\theta$ the fixed point subgroup. Pick a $\theta$-invariant
Borel subgroup $B$. Then there is a natural
map $K\...
- 27.8k
8
votes
1
answer
1k
views
Symmetric tensor product of bosonic/fermionic Hilbert space
Consider two representation of the group $SU(n)$: $Sym^k(\mathbb{C}^n)$ and $\wedge^k\mathbb{C}^n$ ($k\leq n$) and take their symmetric tensor products: $Sym^2(Sym^k(\mathbb{C}^n))$, $Sym^2(\wedge^k\...
- 965
17
votes
0
answers
547
views
Does a symplectic group act on a tensor power of a spin representation?
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$More specifically, let $S_k$ be the spin representation of $\Spin(2k+1)$.
Then is there are action of $\Sp(2r-2)$ on $\bigotimes^{2r}S_k$ ...
- 9,132
5
votes
2
answers
390
views
Dimensions of Jordan blocks associated to representations
Given a linear representation $\rho$ of $SL_n(\mathbb C)$ of finite dimension $m$,
the image $\rho(U)$ of a maximal unipotent Jordan block $U\in SL_n$ decomposes
into generally several Jordan blocks ...
- 17.9k
20
votes
3
answers
840
views
Is there an analogue of the hive model for Littlewood-Richardson coefficients of types $B$, $C$ and $D$?
If $V_\lambda$, $V_\mu$ and $V_\nu$ are irreducible representations of $\operatorname{GL}_n$, the Littlewood-Richardson coefficient $c_{\lambda\mu}^\nu$ denotes the multiplicity of $V_\nu$ in the ...
- 313
2
votes
1
answer
197
views
Polytopes related to the conjugation action of a Lie group on multiple copies of itself?
Let G be a finite dimensional real Lie group. As I understand it, the quotient space of G acting on itself by conjugation is a well studied polytope which can be identified with the fundamental alcove ...
- 27.5k