All Questions
Tagged with algebraic-combinatorics representation-theory
12
questions
2
votes
0
answers
81
views
Restriction of induced representation over a Young subgroup and Littlewood-Richardson coefficients
I'm inexperienced in the representation theory of the symmetric group, so please correct my possible mistakes. Fix $m\leq n$, $G:=S_n$ and $H:=S_m\times S_{n-m}$ as a Young subgroup of $G$. Let $V^{\...
- 4,886
1
vote
0
answers
39
views
Evaluating character functions
Immanants generalize the notion of determinant and permanent, and it is defined as $$Imm_{\lambda}(A)=\sum_{\sigma \in S_n}\chi_{\lambda}(\sigma)\prod_{i=1}^{n}a_{i,\sigma(i)}$$
where $\chi_{\lambda}$ ...
- 125
3
votes
0
answers
99
views
Representation for the Bose-Mesner algebra and its dual.
If you are familiar with the algebra, just skip the following brief introduction, that is fine.
In the study of Bose-Mesner algebra. We know given that the commutative association scheme $\mathfrak{X}...
- 1,366
0
votes
0
answers
107
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Hall Scalar product calculation: $\langle h_3p_6, s_{(5,4)}\rangle$
This is a problem from an old qualifying exam I am reviewing:
Calculate: $\langle h_3p_6, s_{(5,4)}\rangle$ where $\langle \cdot,\cdot \rangle$ is the Hall scalar product defined by $\langle s_\lambda,...
0
votes
0
answers
190
views
Decomposing into irreducible $S_n$ modules, aka Specht modules.
Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$.
I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...
- 2,591
4
votes
0
answers
62
views
Plethysm with Basis?
For any partition $\lambda$ we denote by $S_\lambda$ the corresponding Schur functor. Now consider $\textrm{GL}(\mathbb{C}^n)$ with its natural action on $\mathbb{C}^n$. Using character theory, one ...
- 3,615
1
vote
0
answers
104
views
Crystal operators
Define the operator $s_i$ on tableaux:
Consider letters $i$ and $i + 1$ in row reading word of the tableau.
Successively “bracket” pairs of the form (i + 1, I ).
Left with word of the form $i^r (i +...
- 991
5
votes
2
answers
384
views
$\eta$-value of a partition and its meaning
The $\eta$-value of an integer partition $\lambda = \big( \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 0 \big)$ is defined as
\begin{equation}
\eta \big( \lambda \big) \ := \ \sum_{i=1}^k ...
- 567
9
votes
1
answer
746
views
Young projectors in Fulton and Harris
In Section 4 of Fulton and Harris' book Representation Theory, they give the definition of a Young tableau of shape $\lambda = (\lambda_1,\dots,\lambda_k)$ and then define two subgroups of $S_d$, the ...
- 25.2k
2
votes
0
answers
74
views
Representation theory of the symmetric group
I'm trying to understand the representations of $k\mathfrak{S}_3$, where $k$ is a field of characteristic $2$.
Could someone explain the permutation module $M^{(1^3)}$ has a filtration layers ...
4
votes
2
answers
736
views
Reference request: Representation theory over fields of characteristic zero
Many representation theory textbooks and online resources work with the field of complex numbers or more generally algebraically closed fields of characteristic zero. I am looking for a good textbook ...
- 2,584
4
votes
0
answers
78
views
Is there anywhere some explicit Bruhat decompositions are written down?
Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given.
Also, I calculated the following regarding the Bruhat decomposition of $\...
- 24.9k