scholarly journals Combinatorics of symplectic invariant tensors

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Martin Rubey ◽  
Bruce W. Westbury
Keyword(s):  
Finite Number ◽  
Invariant Theory ◽  
Fundamental Theorem ◽  
Symplectic Group ◽  
Combinatorial Proof ◽  
Second Main Theorem ◽  
Tensor Power ◽  
International Audience ◽  
Cyclic Sieving ◽  
Invariant Tensors

International audience An important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant theory states that all invariants are expressible in terms of a finite number among them, whereas a second main theorem determines the relations between those basic invariants.Here we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic group $Sp(2n)$. Our formulation is completely explicit and provides a very precise link to $(n+1)$-noncrossing perfect matchings, going beyond a dimension count. As a corollary, we obtain an instance of the cyclic sieving phenomenon. Un problème important de la théorie des invariantes est de décrire le sous espace d’une puissance tensorielle d’une représentation invariant à l’action du groupe. Suivant la classique de Weyl, le théorème fondamental premier pour la représentation standard du groupe sympléctique dit que tous les invariants peuvent être exprimés entre un nombre fini d’entre eux. Par ailleurs, un théorème fondamental second détermine les relations entre ces invariants basiques.Ici, nous présentons une preuve transparente d’un théorème fondamental second pour la représentation standard du groupe sympléctique $Sp(2n)$. Notre formulation est complètement explicite et elle fournit un lien très précis avec les couplages parfaits $(n+1)$ -noncroissants, plus précis qu’un dénombrement de la dimension. Comme corollaire nous exhibons un phénomène de crible cyclique.

scholarly journals The first fundamental theorem of invariant theory for the orthosymplectic super group

2018 ◽  
Vol 327 ◽  
pp. 4-24 ◽  
Author(s):  
P. Deligne ◽  
G.I. Lehrer ◽  
R.B. Zhang
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorem

scholarly journals THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

2019 ◽  
pp. 1-25 ◽  
Author(s):  
G. I. LEHRER ◽  
R. B. ZHANG
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorem ◽  
Brauer Algebra ◽  
Tensor Space ◽  
General Linear Supergroup ◽  
Special Cases ◽  
Linear Description ◽  
Tensor Representations ◽  
Capelli Identities ◽  
Fundamental Theorems

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$ . This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$ .

scholarly journals Cyclic sieving for two families of non-crossing graphs

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Svetlana Poznanović
Keyword(s):  
Cyclic Group ◽  
Orbit Structure ◽  
Cyclic Sieving Phenomenon ◽  
Nous Montrons ◽  
International Audience ◽  
Cyclic Sieving ◽  
Cette Action

International audience We prove the cyclic sieving phenomenon for non-crossing forests and non-crossing graphs. More precisely, the cyclic group acts on these graphs naturally by rotation and we show that the orbit structure of this action is encoded by certain polynomials. Our results confirm two conjectures of Alan Guo. Nous prouvons le phénomène de crible cyclique pour les forêts et les graphes sans croisement. Plus précisément, le groupe cyclique agit sur ces graphes naturellement par rotation et nous montrons que la structure d'orbite de cette action est codée par certains polynômes. Nos résultats confirment deux conjectures de Alan Guo.

scholarly journals Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Stirling Numbers ◽  
Algebraic Methods ◽  
Combinatorial Proof ◽  
Nous Obtenons ◽  
Long Cycle ◽  
Bijective Proof ◽  
International Audience

International audience We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size $N+1$. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account. Nous étudions le nombre de permutations $\beta$ de $[N]$ avec $m$ cycles telles que $(1 2 \ldots N) \beta^{-1}$ a un seul cycle. Ces nombres apparaissent en tant que coefficients des monômes linéaires des polynômes de Kerov et de Stanley. À l'aide de méthodes algébriques, D. Zagier a trouvé une connexion inattendue avec les nombres de Stirling de taille $N+1$. Nous présentons ici la première preuve combinatoire de son résultat, en introduisant une nouvelle bijection entre des cartes partitionnées et des arbres épineux. De plus, nous obtenons un résultat plus fin, prenant en compte le type des permutations.

scholarly journals On the density and the structure of the Peirce-like formulae

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Antoine Genitrini ◽  
Jakub Kozik ◽  
Grzegorz Matecki
Keyword(s):  
Finite Number ◽  
Upper Bound ◽  
Large Family ◽  
Partial Answer ◽  
International Audience ◽  
Almost All

International audience Within the language of propositional formulae built on implication and a finite number of variables $k$, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae - Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different $k$ is asymptotically equivalent to the sequence $\frac{1}{ 2 k^2}$. We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by $\frac{c}{ k^3}$ for some constant $c \in \mathbb{R}$. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc - although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to $\frac{1}{ 2 k^2}$.

scholarly journals Cyclic sieving for longest reduced words in the hyperoctahedral group

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
T. K. Petersen ◽  
L. Serrano
Keyword(s):  
Generating Function ◽  
Hyperoctahedral Group ◽  
Cyclic Action ◽  
Orbit Structure ◽  
Cyclic Sieving Phenomenon ◽  
Major Index ◽  
Nous Montrons ◽  
International Audience ◽  
Cyclic Sieving ◽  
Cette Action

International audience We show that the set $R(w_0)$ of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. More specifically, $R(w_0)$ possesses a natural cyclic action given by moving the first letter of a word to the end, and we show that the orbit structure of this action is encoded by the generating function for the major index on $R(w_0)$. Nous montrons que l'ensemble $R(w_0)$ des expressions réduites pour l'élément le plus long du groupe hyperoctaédral présente le phénomène cyclique de tamisage. Plus précisément, $R(w_0)$ possède une action naturelle cyclique donnée par le déplacement de la première lettre d'un mot vers la fin, et nous montrons que la structure d'orbite de cette action est codée par la fonction génératrice pour l'indice majeur sur $R(w_0)$.

scholarly journals The Robinson―Schensted Correspondence and $A_2$-webs

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Matthew Housley ◽  
Heather M. Russell ◽  
Julianna Tymoczko
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Quantum Group ◽  
Classical Limit ◽  
Young Tableaux ◽  
Tensor Power ◽  
Recent Interest ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Graphical Algorithm

International audience The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence.

scholarly journals Combinatorial aspects of pyramids of one-dimensional pieces of fixed integer length

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Bergfinnur Durhuus ◽  
Søren Eilers
Keyword(s):  
Combinatorial Proof ◽  
Average Width ◽  
Fixed Integer ◽  
Bijective Correspondence ◽  
One Dimensional ◽  
International Audience

International audience We consider pyramids made of one-dimensional pieces of fixed integer length $a$ and which may have pairwise overlaps of integer length from $1$ to $a$. We give a combinatorial proof that the number of pyramids of size $m$, i.e., consisting of $m$ pieces, equals $\binom{am-1}{m-1}$ for each $a \geq 2$. This generalises a well known result for $a=2$. A bijective correspondence between so-called right (or left) pyramids and $a$-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids equals $\sqrt{\frac{\pi}{2} a(a-1)m}$.

scholarly journals Cyclic sieving phenomenon in non-crossing connected graphs

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Alan Guo
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Connected Graphs ◽  
Cyclic Sieving Phenomenon ◽  
International Audience ◽  
Cyclic Sieving

International audience A non-crossing connected graph is a connected graph on vertices arranged in a circle such that its edges do not cross. The count for such graphs can be made naturally into a q-binomial generating function. We prove that this generating function exhibits the cyclic sieving phenomenon, as conjectured by S.-P. Eu. Un graphe connexe dont les sommets sont disposés sur un cercle est sans croisement si ses arêtes ne se croisent pas. Nous démontrons une conjecture de S.-P. Eu affirmant que la fonction génératrice q-binomiale dénombrant de tels graphes exhibe le phénomène du crible cyclique.

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