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A question on simultaneous conjugation of permutations

Given $a,b\in S_n$ such that their commutator has at least $n-4$ fixed points, is there an element $z\in S_n$ such that $a^z=a^{-1}$, and $b^z=b^{-1}$? Here $a^z:=z^{-1}az$. Magma says that the ...
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Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls the Nekrasov-...
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A Product Related to Unrestricted Partitions

Start with the product for unrestricted partitions: $(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$ Now replace some of the plus signs with minus signs and expand the product into a series. Is it ...
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Relationship between crystal root operators and usual $e_i, f_i$?

Suppose I am working in a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$. Let $e_1,\dotsc,e_n,f_1,\dotsc,f_n$ denote the usual Chevalley generators of $\mathfrak{g}$. Let $V$ be a highest weight ...
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Non-Boolean Eulerian interval of finite groups

An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following: It ...
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Let $S_u(x)$ be a Schubert polynomial and let $S_v(x;y)$ be a double Schubert polynomial. Then their product can be expressed in terms of the double Schubert polynomials as $$S_u(x)S_v(x;y)=\sum_w{c_{... 0answers 140 views Branching from GL(a+b) to GL(a)\times GL(b) using Gel'fand-Cetlin patterns If one iterates the multiplicity-free branching rule from GL(n) representations (finite-dim, over \mathbb C) to GL(n-1) all the way down to GL(0), one obtains triangular "Gel'fand-Cetlin (or ... 0answers 165 views What is the meaning of the coefficients of the Alekseev-Torossian associator Drinfeld associators became a central object in mathematics and mathematical physics. They appear in deformation quantization, quantum groups, in the proof of the formality of the little disks operad, ... 0answers 244 views Computing the equivariant cohomology class of a Białynicki-Birula cell One of my current research interests is Hessenberg varieties. Briefly, if m_1\le m_2\le \cdots \le m_{n-1} is a weakly increasing sequence of positive integers such that i\le m_i\le n for all i, ... 0answers 340 views Two majs for standard Young tableaux? Let \lambda be a partition of n, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of \lambda, written in English notation, with the numbers 1 ... 0answers 123 views 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, ... 0answers 192 views q-analog of (d/dx) \binom{x}{k}? It is not hard to find easy formulas for the derivative of the function \binom{x}{k}, for instance it is not too hard to see (for k an integer) that \frac{d}{dx} \binom{x}{k} = \sum_{i=1}^k \... 0answers 423 views A relation between intersection and product on Boolean interval of finite groups Let [H,G] be a Boolean interval of finite groups (i.e. the lattice of intermediate subgroups H \subseteq K \subseteq G, is Boolean). For any element K \in [H,G], let K^{\complement} be its ... 0answers 214 views Characterizing n-exceptions of the ring of symmetric polynomials (Also in Mathematics Stack Exchange: https://math.stackexchange.com/questions/2528000/characterizing-n-exceptions-on-the-ring-of-symmetric-polynomials) We say that an homogeneous symmetric polynomial ... 0answers 74 views Littelmann Path model and RSK e and f operators The Littelmann path model defines e_i and f_i operators which correspond in type A to the e_i and f_i operators on semistandard Young tableaux (i.e. since they both are different ways of ... 0answers 171 views Word/Loop in L(\Lambda) Let \mathfrak{g} be a symmetrizable Kac-Moody algebra, with Chevalley generators e_i,f_i (i=1,...,n). Let L(\Lambda) denote the irreducible module with highest weight \Lambda. Let v denote ... 0answers 164 views Is there are good algebraic model of random n-hypergraphs? Suppose F is a finite field and -1 is a square in F. Let E be the binary relation on F where (a,b) \in E iff a - b is a square. Then (F,E) is called a Paley graph. Paley graphs are ... 0answers 296 views Coefficients in expansion of a classical symmetric polynomial If we expand $$P_3(x_1,\ldots, x_n):=\Pi_{1\leq i<j<k\leq n} (x_i+x_j+x_k),$$ then P_3 = \sum_\alpha \sum_{\mathcal{O}(\alpha)} c_\alpha x_1^{\... 0answers 194 views Can Matsumoto's theorem for the symmetric group be proved using a monovariant? This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions: Let n be a nonnegative integer. ... 0answers 66 views Form on symmetric functions and their q,t- analogues [Notations are as in Macdonald's Symmetric Functions and Hall Polynomials] The space of symmetric functions \Lambda_{\mathbb{Q}} has a bilinear form defined by  (p_\lambda, p_\mu)= z_\lambda \... 0answers 61 views Embedding the Mészáros subdivision algebra in an Orlik-Terao localization The following is an open question (Question 4.1) from my paper t-Unique Reductions for Mészáros's Subdivision Algebra (published version in SIGMA 2018, and slightly updated preprint version with ... 0answers 213 views 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^{\... 0answers 826 views Internal tensor product of strict polynomial functors: is there a more explicit definition? In the paper Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors, arXiv:1203.0311v4, Krause defines something that he calls an "internal tensor product" on the category of ... 0answers 137 views Is the Euler characteristic of a subfactor planar algebra, nonzero? Let \mathcal{P} be an irreducible subfactor planar algebra and \mu the Möbius function of its biprojection lattice [e_1,id]. Then the Euler characteristic of \mathcal{P} is defined as follows: ... 0answers 460 views 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 ... 0answers 146 views Counting square zero forms over finite fields Let p be an odd prime and let R=\Lambda_{\mathbb{F}_p}[x_1,\dots,x_n] be the exterior algebra on n generators over the finite field with p elements. This is a graded-commutative ring. Is ... 0answers 209 views Anti-arithmetic product of symmetric functions: (why) is it integral? This is an analogue of MathOverflow question #138148. Indeed it is so analogous that I wrote the following by copypasting said question and making the necessary changes. For every commutative ring A... 0answers 230 views Counting perfect matchings with integrals Has anyone used the Joni-Rota-Godsil integral formula (see details below) to count perfect matchings of square-grid graphs, Aztec diamond graphs, hexagon-honeycomb graphs, etc.? (Or even just to ... 0answers 344 views Littlewood-Richardson rule for Schubert polynomials What is the current state of the problem of finding a combinatorial rule for multiplying two Schubert polynomials? Is the problem still open? 0answers 85 views Expressing symmetric function in power-sum basis I am trying to prove the following identity \prod_{i=1}^{m}(1-x_{i}z)^{-u}\prod_{j=1}^{m}(1-y_{i}z)^{-v} \prod_{i=1}^{n}\prod_{j=1}^{n}(1-(x_i +y_j)z)^{-w}\\ = \sum_{\lambda, \mu}c_{\... 0answers 142 views Does every finite lattice embed into a finite Eulerian lattice? A finite Boolean lattice is a lattice isomorphic to the subset lattice of a finite set. Every Boolean lattice is Eulerian, namely, a graded lattice L such that \mu(a,b) = (-1)^{|b|-|a|} for all a,... 0answers 105 views A combinatorial proof of an identity of partitions (Macdonald I.5) This is a statement from Symmetric Functions and Hall Polynomials by Macdonald: \sum_{x\in \lambda} (h(x)^2-c(x)^2)=|\lambda|^2 where \lambda denotes a partition or a Young diagram, and for each ... 0answers 228 views Generalization of a theorem of Øystein Ore in group theory: the infinite case This post is the infinite version of this one, and is motivated by an exchange with Carmela Musella and Maria De Falco. We are interested in relative versions of the following Ore's theorem and ... 0answers 171 views Local structure of non-normal toric varieties---possible mistake in "Discriminants, Resultants and Multidimensional Determinants" I believe I may have a counterexample to Theorem 5.3.1 on page 179 from the book book Discriminants, Resultants and Multidimensional Determinants by Gel'fand, Kapranov, and Zelevinsky. To summarize ... 0answers 91 views Rationality of power series whose coefficients are the ranks of a sequence of matrices Recently, I stumbled several times about the problem to decide whether a certain formal power series$$ f = \sum_{n=0}^\infty d_n T^n \in \mathbb{Q}[\![T]\!]$$is actually a rational function, where ... 0answers 781 views 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}\... 0answers 157 views Chern character of Schubert structure sheaf Let X_\lambda \subset Gr = Gr(k,n) be a Schubert variety in the Grassmannian and \mathrm{ch} : K_0(Gr) \otimes \mathbb{Q} \to A^\bullet(Gr) \otimes \mathbb{Q} the Chern character isomorphism. Is ... 0answers 332 views Can the Littlewood-Richardson cone be used for combinatorial optimization? The Littlewood-Richardson cone LR_{n, k} consists of all k-tuples (a_1, a_2, \dots, a_k) of real n-vectors with monotonically decreasing entries such that there exist k n \times n-... 0answers 104 views Polynomiality of the inverse of equally weighted Varchenko matrices attached to hyperplane arrangements Let d\in \mathbb{Z}_{\ge 1}, let \sigma = (H_i)_{i\in \mathcal{I}} be a finite hyperplane arrangement in \mathbb{R}^d, where H_i\subset \mathbb{R}^d is a hyperplane for i\in \mathcal{I} (the ... 0answers 100 views Can equality of chromatic symmetric functions of two trees be checked in polynomial time? Stanley defined chromatic symmetric functions (CSF) in 1995 (Advances in Math) where he conjectured that trees can be distinguished by their CSF. However, tree isomorphism is decidable in P (... 0answers 72 views Subalgebras of a polynomial ring carved out by (families of) coefficient equalities Let \mathbf{k} be a field, and let P=\mathbf{k}\left[ x_{1},x_{2} ,\ldots,x_{n}\right]  be a polynomial ring over \mathbf{k} in n variables x_{1},x_{2},\ldots,x_{n}. Alternatively, P can ... 0answers 117 views Number of adjoint orbits containing a (0,1)-matrix Motivated by this question, what can be said about the number f(n) of adjoint orbits of \mathrm{Mat}(n,\mathbb{C}) (the ring of all n\times n complex matrices) that contain a (0,1)-matrix? ... 0answers 79 views Decomposition of Schur modules over the orthogonal group Let V=\mathbb{R}^n and O(n) the orthogonal group acting with its standard action on V. Now for any partition \lambda we have the Schur module S_\lambda V which is a representation of O(n). ... 0answers 775 views Frobenius formula I know two formulas by the name of Frobenius. The first one computes the number$$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$where ... 0answers 104 views Shifted schur function and holonomic Now let us denote by \Lambda^{*}(n) the algebra of polynomials in x_{1},\ldots,x_{n} that become symmetric in new variables$$ x_{i}'=x_{i}-i+c, \ i \in 1,\ldots,n. Here c is a arbitrary fixed ...
Consider a $n$ by $n$ by $n$ grid represented by the set of $3$-uples $S=\{1,2,\dots, n\}^3$. A line (resp. slice) of $S$ is a subset of cardinal $n$ (resp. $n^2$) where two components (resp. one ...