Questions tagged [young-tableaux]
For standard Young tableaux, semistandard Young tableaux, and other related two-dimensional arrays of numbers like plane partitions. Including their combinatorial theory and their application in representation theory and algebraic geometry.
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Determinantal formula for plane partitions of shifted shape
For $\lambda = (\lambda_1,\ldots,\lambda_{\ell})$ a partition, a (weak) plane partition of shape $\lambda$ is a filling of the Young diagram of $\lambda$ with nonnegative integers such that entries ...
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Tensoring irreducible $B$-series representations/ Type B Littlewood-Richardson
When tensoring finite dimensional representations of the Lie algebra ${\frak sl}_n$, we have an explicit algorithm given in terms of Young diagrams. See Section 4 of this paper.
Do there exist ...
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Restricted Cauchy identity
Is there some reference for sums like:
$$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)t^{|\nu|}$$
$$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)\cdot|\nu|$$
(summation ...
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Constructing a centrally primitive idempotent in the group algebra of the symmetric group
Consider the group algebra of the symmetric group $ \mathbb{C} S_k$.
Given some Young tableau $T$ of shape $\lambda$, let $a_{\lambda,T}$ and $b_{\lambda,T}$ be the row symmetrizer and column ...
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Iterated derivative and rectangular standard Young tableaux
We first make a few definitions, seemingly out of the blue (they are introduced/defined in this paper).
Let $F^0_{a}(z) = (1-z)^{-1}$ and define recursively
$$
F^{k+1}_{a}(z) = z^{a-1} \frac{d^a}{dz^...
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Decomposing tensor powers of the fundamental representation of exceptional Lie algebras
For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, ...
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About $K$-rectification of increasing tableaux
Let $T$ be a standard Young tableaux on $[n]$. Denote the RSK algorithm $\text{RSK}(w)=(P(T),Q(T))$ for $w\in\mathfrak{S}_n$, where $P(T)$ is the Schencted insertion tableaux.
For $1\leq i\leq j\leq n$...
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Intersection of components in Springer fibre of type A
From the standard results on Springer fibers of type A, we know that given a Springer fiber, say $\mathcal{B}_\lambda,$ its irreducible components are all equidimensional and parametrized by standard ...
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Coincidences between average Catalan tableaux
There are Catalan number $C_n$ of standard Young tableaux of shape $(n,n)$, which we view as $2\times n$ matrices. Denote by $P_n$ the average of these matrices:
$$
P_n \, := \, \frac{1}{C_n} \, \...
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Continuous analogues of Schützenberger promotion
Has anyone studied continuous analogues of Schützenberger promotion, and in particular, a flow on (a suitable subset of) the order polytope of a poset?
Here’s what I have in mind: Given a poset $P$, ...
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Major index generating polynomial for border-strip tableaux
The Question in its original form has been answered, but there is a follow-up, see the end of the post.
A border-strip is a skew Young diagram that does not contain a $2 \times 2$-box. A border-strip ...
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Generating function for lattice paths making aribitrary (i,j)-up-right move in one step and fitting rectangular (m,n)?
There is the following beautiful formula (see Qiaochu Yuan excellent blog):
$$ \sum_{\lambda \in Young~diagrams~fitting~rectangle~m~n} q^{Box~count(="area~under~the~curve")~of~\lambda} = \binom{n+m}{...
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To find $\dim(D^{\mu})=\dim\frac{S^{\mu}}{S^{\mu}\cap S^{\mu\perp}}$
Let $S_6$ be the symmetric group of degree $6$ and $F$ be any finite field of characteristic $2.$ Then $2$-regular partition of $6$ are $(5,1)$, $(4,2)$ and $(3,2,1)$ . I have to find $$\dim(D^{\mu})=\...
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Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials
$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}_n$ be the symmetric group, $\SYT_n$ be the set of standard young tableaux of size $n$.
For $u\in \...
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How to represent the even signed permutations by Young tableaux?
The well-known RSK correspondence established the connection between table pair (P,Q) and the permutations in symmetry group Sn(Coxeter group of type A). Also, there is a similar correspondence for ...
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Bijection between noncrossing matchings on $2b$ points and Standard Young Tableaux of size $2 \times b$
I'm currently reading a review article called Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance by Jessica Striker. In this article, Striker writes that there is a ''nice'' ...
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Nekrasov Partition Function: $F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q})$ analytic at $\epsilon_1 = \epsilon_2 = 0$?
Nakajima & Yoshioka [1] showed that
\begin{equation}
F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q}) = \sum_{n = 1}^\infty \mathbf{q}^nF^{inst}_n(\epsilon_1,\epsilon_2,\mathbf{a}) := \...
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Is the Ford-Fulkerson algorithm a tropical rational function?
The Ford-Fulkerson algorithm
Let me recall the standard scenario of flow optimization (for integer flows at least):
Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
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A dimension formula for generalised symmetric powers of the natural module
I need a reference for the following well-known statement - does anyone know one?
Let $\mu$ a partition of $n$ into at most $d$ parts.
We let
$${\rm Sym}^\mu(\Bbbk^d)={\rm Sym}^{\mu_1}(\Bbbk^d) \...
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Tabloid Construction of Permutation Representation of Hyperoctahedral Group
For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\...
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Orthogonal basis for decomposition of induced representation of derangements
Background
Let $V_n$ be the $\mathbb{C}$-module spanned by the set of derangements (permutations with no fixed points) inside the group ring of $S_n$. We make $V_n$ into a $\mathbb{C}S_n$-module ...
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Dimension of irreducible representation associated to a Young tableau
This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.
Suppose that $\lambda_1 \geq \lambda_2 \geq \dots \lambda_k$ and $\rho$ be ...
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Skew character with hooks
I have asked this in MSE 8 days ago, even offered a bounty, and got nothing, so will try here.
I would like to understand the value of the skew characters of the symmetric group, $\chi_{\lambda/\mu}$ ...
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Natural maps between Schur functors: understanding the image
Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map
$$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$
Let $[\Lambda^2 V]...
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Multidimensional hook length formula
A well-known hook length formula states that the number of ways to arrange the elements of $[n]$ in a Young tableau with $n$ cells so that all columns and rows are increasing is $\frac{n!}{\prod_c h(c)...
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Height growth for randomly falling Tetris like blocks ? What if Young diagrams are falling down?
Question: How the maximal height grows for random Tetris like blocks falling down ? Numeric simulation (see below)
shows leading term is linear with some constant
depending on shapes of blocks ...
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Is there winning strategy in Tetris ? What if Young diagrams are falling?
Question 1
Is there a winning strategy (algorithm to play infinitely) in Tetris,
or is there a sequence of bricks which is impossible to pack without holes?
Consider generalized Tetris with Young ...
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Categorifying the Cauchy kernel as a filtration of $\operatorname*{Sym}\left( F\otimes G\right) $ over any commutative ring
Question 1 (short version). Let $R$ be a commutative ring with unity. Let
$F$ and $G$ be two $R$-modules. Let $n\in\mathbb{N}$. Is it true that the
$n$-th symmetric power $\operatorname*{Sym}\...
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"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 ...
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Hook-content polynomial 2
Recently I have proven the following identity
\begin{align}
\sum_{\lambda\in \text{different hook of size d}} \frac{1}{d!} (-1)^{ht(\lambda)-1} \, \dim \lambda \, \prod_{\Box \in \lambda} \frac{1}{1-...
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Induced representation of a Young subgroup
This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.
Suppose that $n=k+l+r$ where $k\geq l\geq r\geq 0$. Let $G$ be the symmetric ...
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Bijective proof of an identity involving number of standard Young tableaux and semistandard tableaux
Question. Can you find a bijective proof of the identity
$$ \operatorname{dim}(S^{\lambda} \mathbb{C}^m)\ \operatorname{dim}(S^{\lambda'} \mathbb{C}^n) \ f^{n^m}
= \dim \Lambda^p (\mathbb{C}^m \...
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Littlewood-Richardson sequences and Littlewood-Richardson coefficients
I'm looking for a proof or a reference for the following statement, I give the definitions below:
There exists a Littlewood-Richardson sequence of type $(\alpha, \beta, \lambda)$ if and only if $...
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Proportion of partitions in a rectangle
Let $ k, n, r \geqslant 1 $ be integers. Let $ \lambda $ be a partition of $r$, what we denote by $ \lambda \vdash r $.
I would like a lower and an upper bound for the following quantity, for all $ ...
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What can be said of a $6$-core Young diagram whose $2$-and $3$-cores are empty
Let $\lambda$ be a partition. Suppose that $\lambda$ is both $2$- and $3$-decomposable, in the sense that $\lambda$ admits a total decomposition by both $2$-rim hooks (aka dominos) and $3$-rim hooks. ...
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"quantum" symmetric plane partitions beget alternating sign matrices?
The "quantum" version qTSPP of the number of totally symmetric plane partitions, contained in the cube $[0,n]^3$, is enumerated by
$$f_n(q):=\prod_{j=1}^n\prod_{k=1}^j\prod_{\ell=1}^k\frac{1-...
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Symmetric group action on Young Tableaux
Let $S_n$ be the symmetric group and let $\lambda$ be a partition of $n$ and let $S$ be the set of all standard young tableaux of shape $\lambda$. Is there a well defined action of $S_n$ on the set $S$...
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Young Symmetrizer and Exterior Products, such as $S_{(2,1)}V = Ker(\Lambda^2V \otimes V \to \Lambda^3V )$
Background: Young symmetrizer $c_\lambda$ gives an explicit description of Schur module $S_\lambda V$, which is also the kernel of maps between exterior products (as in Fulton & Harris).
Example:...
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Correspondence between $SBT (n)$ and $W(B_n)$
Let $W(B_n)$ be a Weyl group of type $B_n$ and $SBT (n)$ the set of standard bitableaux of size $n$. Similar to Robinson-Schensted correspondences, I know that there exists a map $W(B_n) \to SBT (n) \...
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Identities satisfied by the image of the Young symmetrizer
Consider a partition $\lambda=(r_1,\ldots,r_k)$ of an integer $n$ and the corresponding Young diagram with rows of length $r_1,\ldots,r_k$ (hence ordered in non-increasing order). Counting the column ...
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Counting the orbits of a set of tabloids under the action of a Young subgroup
Let $\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k)$ and $\mu = (\mu_1 \geq \mu_2 \geq \cdots \geq \mu_\ell)$ be partitions of a positive integer $n$. As in Fulton's book on Young ...
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On an asymptotic formula of Keating and Snaith involving the Riemann zeta function
Keating and Snaith have a famous conjecture on the asymptotics of the
integral $\int_0^T |\zeta(\frac 12+it)|^{2k}\, dt$, where $\zeta$
denotes the Riemann zeta function. See page 510 of the book ...
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Does flipping Young diagrams has anything to do with Fourier?
Here's a chaser to this question.
Recall the proof that the number of partition of an integer $n$ into at most $k$ addends is the same as the number of partition of an integer $n$ into integers no ...
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Inequality for hook numbers in Young diagrams
Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$, define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
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Characterizing the RSK corespondance
The Robinson-Schensted-Knuth correspondence is an algorithm which takes as input a word $w$ on the alphabet $\{1,\dots,n\}$ of length $k$ and returns a pair of a tableau $P(w)$ and a standard tableau $...
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"Diagonalizing" Littlewood-Richardson coefficients
Let's consider the Littlewood-Richardson coefficients $c^{\lambda}_{\mu \nu}$ so that
\begin{equation}
V_\mu \otimes V_\nu = \bigoplus_\lambda V_\lambda^{\oplus c^{\lambda}_{\mu \nu}}
\end{equation}
...
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The properties of Pos
Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as:
$$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$
When given $n\in\mathbb{N}$, this function gives the 'position' of $...
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1
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Schubert varieties and Young diagrams
In his book Young Tableaux, Fulton asserts, in Exercise 9.4.18 on p. 152, that the Schubert variety $\Omega_{\lambda}$ is defined by the conditions $\text{dim}(V \cap F_{n+i- \lambda_{i}}) \geq i$ for ...
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Tableaux with limited rows and complementary skew shapes
Given a partition $\mu=(\mu_1,\mu_2...,\mu_d)$, define $\bar\mu=(\mu_1-\mu_d,\mu_1-\mu_{d-1},...,\mu_1-\mu_2,0)$, the complementary shape in the $d\times \mu_1$ rectangle. Then the number of skew ...
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Confusion with proof of Pieri's Formula
In Manivel's Symmetric Functions, Schubert Polynomials, and Degeneracy Loci, I am confused with some parts of his proof of Pieri's Formula. It is given as Pieri's Formula $3.2.8$ (p. $109$):
If $\...
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