ترغب بنشر مسار تعليمي؟ اضغط هنا

Crystals, semistandard tableaux and cyclic sieving phenomenon

67   0   0.0 ( 0 )
 نشر من قبل Euiyong Park
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

In this paper, we study a new cyclic sieving phenomenon on the set $mathsf{SST}_n(lambda)$ of semistandard Young tableaux with the cyclic action $mathsf{c}$ arising from its $U_q(mathfrak{sl}_n)$-crystal structure. We prove that if $lambda$ is a Young diagram with $ell(lambda) < n$ and $gcd( n, |lambda| )=1$, then the triple $left( mathsf{SST}_n(lambda), mathsf{C}, q^{- kappa(lambda)} s_lambda(1,q, ldots, q^{n-1}) right) $ exhibits the cyclic sieving phenomenon, where $mathsf{C}$ is the cyclic group generated by $mathsf{c}$. We further investigate a connection between $mathsf{c}$ and the promotion $mathsf{pr}$ and show the bicyclic sieving phenomenon given by $mathsf{c}$ and $mathsf{pr}^n$ for hook shape.



قيم البحث

اقرأ أيضاً

In this paper, we compute explicitly the $q$-dimensions of highest weight crystals modulo $q^n-1$ for a quantum group of arbitrary finite type under certain assumption, and interpret the modulo computations in terms of the cyclic sieving phenomenon. This interpretation gives an affirmative answer to the conjecture by Alexandersson and Amini. As an application, under the assumption that $lambda$ is a partition of length $<m$ and there exists a fixed point in $mathsf{SST}_m(lambda)$ under the action $mathsf{c}$ arising from the crystal structure, we show that the triple $(mathsf{SST}_m(lambda), langle mathsf{c} rangle, mathsf{s}_{lambda}(1,q,q^2, ldots, q^{m-1}))$ exhibits the cycle sieving phenomenon if and only if $lambda$ is of the form $((am)^{b})$, where either $b=1$ or $m-1$. Moreover, in this case, we give an explicit formula to compute the number of all orbits of size $d$ for each divisor $d$ of $n$.
84 - Seung Jin Lee 2019
In 1976, King defined certain tableaux model, called King tableaux in this paper, counting weight multiplicities of irreducible representation of the symplectic group $Sp(2m)$ for a given dominant weight. Since Kashiwara defined crystals, it is an op en problem to provide a crystal structure on King tableaux. In this paper, we present crystal structures on King tableaux and semistandard oscillating tableaux. The semistandard oscillating tableaux naturally appear as $Q$-tableaux in the symplectic version of RSK algorithms. As an application, we discuss Littlewood-Richardson coefficients for $Sp(2m)$ in terms of semistandard oscillating tableaux.
100 - Sen-Peng Eu , Tung-Shan Fu 2006
The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridges $q=-1$ phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization of the cluster complexes found by Fomin and Zelevinsky. In this paper, the faces of various dimensions of the generalized cluster complexes in type $A_n$, $B_n$, $D_n$, and $I_2(a)$ are shown to exhibit the cyclic sieving phenomenon under a cyclic group action. For the cluster complexes of exceptional type $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, and $H_4$, a verification for such a phenomenon on their maximal faces is given.
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).
By considering the specialisation $s_{lambda}(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $lambda$ in terms of two properties of the boxes in the diagram f or $lambda$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $lambda$-tableaux.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا