# Tagged Questions

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### Representation of Permutation group: What is the isormorphism between the equivalent descriptions, (1)using schur functions and (2)abstract tensors

I assume that the two ways of describing the representation of the permutation group using abstract tensors (perhaps more naive but widely used in physics) and schur functions are equivalent. If ...
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### Is the reduced plethysm (restricted to 2-columns in Young tableaux) of this Schur funtion known $\mathbb S_{3^1}(\mathbb S_{1^p})$?

I am working on a physical problem, where I need to compute the "reduced plethysm" that is all the irreducibles characterised by the Young tableaux of 2 columns or less. The plethysm problem I want to ...
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### A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...
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We say that a multilinear polynomial $P(x_1,\ldots,x_n)$ in $n$ commuting variables over $\mathbb{R}$ has zero trace if $$\frac{d}{dt} P(t,\ldots,t) = 0.$$ Equivalently, $$\left(\sum_{i=1}^n ... 0answers 119 views ### characterization of all periodic tiling of a simple set of Wang Tile Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set. Now, I wish to characterize all the periodic tilings of this set (better if they are ... 1answer 198 views ### Question about a proof in Graham and Lehrer's “Cellular algebras” I'm sorry if this question is too basic for MO. I'm reading a paper by Graham and Lehrer "Cellular algebras" and have trouble understanding one step in a proof of a crucial theorem. I suppose that the ... 0answers 356 views ### Combinatorics of Quantum Schubert Polynomials Let S_n be the symmetric group. Let s_i denote the adjacent transposition (i \ i+1). For any permutation w\in S_n, an expression w=s_{i_1}s_{i_2}\cdots s_{i_p} of minimal possible length is ... 0answers 176 views ### Degree of a cone over the set of rank r n\times n matrices Let X_r\subset Mat_{n\times n}(C) denote the variety of rank at most r matrices, set k=n-r and assume n\geq k^2-1. Consider the cone over X_r with vertex spanned by the first k^2-1 entries ... 1answer 507 views ### Counting representations of k[x,y] when k is finite \newcommand{\GFq}{\mathbf F_q} Let r_n(q) denote the number of isomorphism classes of n-dimensional modules of the \GFq-algebra \GFq[x,y]. Is it known whether there exists a polynomial ... 2answers 417 views ### calculating Littlewood-Richardson coefficients It is known that if \alpha,\beta,\gamma are three partitions then the Littlewood-Richardson coefficient c_{\alpha \beta}^{\gamma} is positive when the triple (\alpha,\beta,\gamma ) occurs as ... 1answer 261 views ### Iterated Pieri's rule, Schur functors and intersection of subrepresentations Let \lambda and \mu be two Young diagrams, such that \lambda can be obtained from \mu by extending one single column with additional b boxes. Let \Sigma^\lambda U and \Sigma^\mu U denote ... 0answers 151 views ### Is there any natural construction for the irreducible representations of G\wr S_n? For S_n, one can construct all the irreducible representations through the young diagrams. Is there any natural construction for the irreducible representation of G\wr S_n (G is a finite group)? 1answer 134 views ### Help identify this generalized sign of real representations Let V be a real representation of a finite group G. Define \mathbb Z[I]_{I\leq G} to be the ring over the integers generated by subgroups of G with multiplication corresponding to ... 1answer 532 views ### Conjectural identities for Young symmetrizers and Young-Jucys-Murphy elements The following questions I have found in my own notes from about 3 years ago. Unfortunately, I lost much of the context; I believe I made these conjectures reading Okounkov-Vershik, arXiv:0503040v3, ... 0answers 135 views ### Is there an Arctic Circle phenomenon for Amman-Beenker tilings? I found some slides on tilings and one of them pertained to Amman-Beenker tilings. It looks like there is an Arctic Circle phenomenon similar to that for dominos or lozenges. Is there any ... 1answer 496 views ### On Applications of Murnaghan Nakayama Rule This question is crossposted at math.stackexchange here and may be beyond the usual scope of the site. The question is located below. In short, I am looking for an accessible explanation of the ... 0answers 85 views ### Differences of Numbers of Helicity States in 4-dimensional Strings The question whether the states in D=2m + 2 dimensional string theory, which carry a representation of SO(2m), span spaces which carry representations of SO(2m+1) seems hopelessly complicated. ... 1answer 93 views ### Previous derivation of a combinatorial formula for the fundamental representations of A_{n-1} Consider the simple Lie algebra A_{n-1} over the complex numbers. I refer to A_{n-1} rather than A_n for reasons that I try to make clearer below. The following facts are no doubt well known, ... 0answers 111 views ### Semi-Standard Young Tableaux: Do Diagrams for O(2m) combine to Diagrams from O(2m+1)? Let n_\lambda^K be the number all semi-standard Young tableaux of size K with Ferrers diagrams diagram \lambda (i.e. the number of all fillings of \lambda with natural numbers with weakly ... 2answers 299 views ### Symmetric powers of Schur polynomials I apologize if this question is trivial, but a couple of days of searching for necessary routines have led me here. Does there exist software to compute symmetric powers of Schur polynomials? ... 2answers 284 views ### Temperley-Lieb algebras for other Weyl groups? The Temperley-Lieb algebra has the same generators as the S_n group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen ... 1answer 194 views ### Combinatorics of index sets multiplicities in characters of symmetric groups Hi everyone. I'm pondering the following question: I have a Coxeter group (W,S) of type A_{n-1}, i.e. the symmetric group W=Sym(n) with the neighbour transpositions as generating set S=\lbrace ... 0answers 411 views ### Reference/quote request: “All of combinatorics is the representation theory of S_n” I think I remember reading somewhere a glib (or is it deep?) quote, perhaps due to Rota?, which was something like "All of combinatorics is essentially [or can be reduced to?] the representation ... 0answers 139 views ### Summing Characters of the Symmetric Group over Derangements (Enumerative Combinatorics: Vol. II Ex. 7.63) The following exercise is out of Stanley's Enumerative Combinatorics: Vol. II (Ex. 7.63): For \lambda \vdash n define d_\lambda = \sum_{w \in D_n} \chi^\lambda(w) where D_n is the set of all ... 3answers 539 views ### Is there a list of Kazhdan-lusztig polynomials? When studying the combinatorics and representation of Coxeter groups, I often find it irksome to compute KL- and R- polynomials from scratch on Maple or Sage. The time it consumes to generate a ... 1answer 407 views ### Hilbert Matrix and Approximation Theory I was reading about the Hilbert matrix and Cauchy determinants: $\det \left[ \frac{1}{i+j-1} \right]_{i,j}$ By guessing where this determinant is 0 or \infty we can guess the right formula. ... 0answers 91 views ### Next smallest dimension of Specht Module after (n), (1^n), (n-1,1) and (2,1^{n-2}) In representation theory of S_n, we know that for n \geq 9, the only Specht modules S^\alpha of dimension f^\alpha < {n-1 \choose 2} - 1 are: S^{(n)} and S^{(1^n)} with dimension 1, ... 0answers 158 views ### Eigenvalues of “modified” Johnson scheme via the representation theory of the symmetric group I am interested in eigenvalues of the following association scheme, which somewhat resembles the Johnson scheme. Let n and k\leq n be positive integers. The n!/(n-k)! vertices of the scheme ... 1answer 173 views ### A question on Lusztig's `graph with automorphism' construction? Using the notion of a graph with compatible automorphism, Lusztig constructs all symmetrizable Cartan data (i.e. Cartan matrices A for which there is a diagonal matrix ... 1answer 178 views ### PROPs representations, free module analog Ordinary operad with one ouput can be obviously regarded as free module on itself. Is there are analogous construction for operad with many outputs (PROP)? This must be difficult question, but what is ... 1answer 350 views ### Known decomposition of \bigwedge^k Sym^d \mathbb C^n in special cases? Let V = \mathbb C^n. Consider the plethysm \bigwedge^k Sym^d V as a representation of GL(V). In what special cases (e.g., for what k, d, and n) is this representation's decomposition into ... 0answers 130 views ### Harmonic analysis and non-symmetric Macdonald polynomials? I have recently been reading a lot about Macdonald polynomials, the symmetric and the non-symmetric ones. One thing that strikes me is that the symmetric Macdonald polynomials admit a positive theory, ... 4answers 618 views ### Largest permutation group without 2-cycles or 3-cycles The largest permutation group without 2-cycles is A_n, which has size n!/2. I think the largest permutation group without 2-cycles or 3-cycles is much smaller, but I can't figure out if it should ... 1answer 1k views ### Order of products of elements in symmetric groups Let n \in \mathbb{N}. Is it true that for any a, b, c \in \mathbb{N} satisfying 1 < a, b, c \leq n-2 the symmetric group {\rm S}_n has elements of order a and b whose product has order ... 1answer 406 views ### Littlewood Richardson rule and seminormal basis of Specht modules Background Seminormal Basis of Specht modules of \mathfrak{S}_n Let \lambda be a partition of n. A \lambda-tableau is a bijection \mathfrak{t}:\lambda \to \{1,2,...,n\}. We say a ... 2answers 505 views ### Is there formula name and proof for this theorem ? [closed] The formula answers: how many tuples (\sigma_1,\sigma_2,...,\sigma_n) of elements of a given group G such that (1) \sigma_i\in C_i , where C_i stands for conjugacy class. (2) ... 0answers 247 views ### det(A)det(B) = det(AB+correction), Capelli identities, “factorzied” representation of gl_n Context: some probably know that there are Capelli identities which state det(A)det(B) = det(AB+correction) for some matrices with non-commuting elements, they go back to 19-th century, but also ... 0answers 419 views ### Cauchy matrices with elementary symmetric polynomials \newcommand{\vx}{\mathbf{x}} Let e_k(\vx) denote the elementary symmetric polynomial, defined for k=0,1,\ldots,n over a vector \vx=(x_1,\ldots,x_n) by \begin{equation*} e_k(\vx) := \sum_{1 ... 1answer 868 views ### Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials? In their seminal 1979 paper here, Kazhdan and Lusztig studied an arbitrary Coxeter group (W,S) and the corresponding Iwahori-Hecke algebra. In particular they showed how to pass from a standard ... 2answers 421 views ### elements in the weyl group Let W be the Weyl a group of a semisimple simply connected group over C. Let I={1,...,r} the set of simple roots. For w\in W, I denote by supp(w) the subset of I corresponding to the simple ... 1answer 228 views ### Identity involving partitions coming from representations of alternating groups It is not difficult to show that the number of conjugacy classes in the alternating group A_n is given by classes in the alternating group = no. of even partitions + no. of self-transpose ... 1answer 302 views ### On q-Demazure operators Setup Let G be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic with Borel subgroup B. Let \Lambda denote the weight lattice of G; we write elements ... 3answers 1k views ### Tensor powers of the standard representation Consider V_{(n-1, 1)}, the n-1 dimensional irreducible representation of S_n, i.e. the "standard" or "defining" representation. Is there a nice formula for how the k-th tensor power of ... 0answers 260 views ### Chern Classes of Exterior Products of a vector bundle. This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down c(\wedge^k V) i.e. the individual summands in terms of the individual summands of ... 1answer 336 views ### Explicit method to compute Macdonald/Koornwinder functions I'd like to compute explicitly symmetric Macdonald functions associated to arbitrary (possibly non-reduced) root systems, using Computer Algebra System. Unfortunately Sage seems to only implement ... 1answer 510 views ### Reconstruction Conjecture holds for Directed Acyclic Graphs? Wikipedia's article on the Reconstruction Conjecture mentions that the conjecture is false for digraphs, and refers to two papers by Stockmeyer. As far as I can see, none of the counter-examples in ... 2answers 407 views ### Can one recognize this symmetric function? \newcommand{\lm}{\lambda} \newcommand{\bR}{\mathbb{R}} Let m be an integer >1. Define$$ I_m:\bR^m\to \bR,\;\; I_m(\lm_1,\dotsc, \lm_m)=\int_{S^{m-1}}\exp\Bigl(-\sum_{j=1}^m ...
Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers ...