Crystal Graphs and qq-Analogues of Weight Multiplicities for the Root System AnA_n

We give an expression of the $q$-analogues of the multiplicities of weights in irreducible \sl_{n+1}-modules in terms of the geometry of the crystal graph attached to the corresponding U_q(\sl_{n+1})-modules. As an application, we describe multivariate polynomial analogues of the multiplicities of the zero weight, refining Kostant's generalized exponents.Comment: LaTeX file with epic, eepic pictures, 17 pages, November 1994, to appear in Lett. Math. Phy

Partition Analysis and Symmetrizing Operators

Using a symmetrizing operator, we give a new expression for the Omega operator used by MacMahon in Partition Analysis, and given a new life by Andrews and his coworkers. Our result is stated in terms of Schur functions.Comment: 5 page

q-Identities from Lagrange and Newton Interpolation

Combining Newton and Lagrange interpolation, we give $q$-identities which generalize results of Van Hamme, Uchimura, Dilcher and Prodinger

Enumeration of bigrassmannian permutations below a permutation in Bruhat order

In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schutzenberger and Reading.Comment: 7 pages

Schur Q-functions and degeneracy locus formulas for morphisms with symmetries

We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle, and is based on a certain push-forward formula for these polynomials in a Grassmann bundle.Comment: 22 pages, AMSTEX, misprints corrected, exposition improved. to appear in the Proceedings of Intersection Theory Conference in Bologna, "Progress in Mathematics", Birkhause

Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A

We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out to be expressible only in terms of: - Schensted's classic bumping procedure, - the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to describe, acting on cylindric multipartitions. We explain how this can be seen as an analogue of the bumping algorithm for affine type $A$. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space

Origin and demographic history of the endemic Taiwan spruce (Picea morrisonicola)

Taiwan spruce (Picea morrisonicola) is a vulnerable conifer species endemic to the island of Taiwan. A warming climate and competition from subtropical tree species has limited the range of Taiwan spruce to the higher altitudes of the island. Using seeds sampled from an area in the central mountain range of Taiwan, 15 nuclear loci were sequenced in order to measure genetic variation and to assess the long-term genetic stability of the species. Genetic diversity is low and comparable to other spruce species with limited ranges such as Picea breweriana, Picea chihuahuana, and Picea schrenkiana. Importantly, analysis using approximate Bayesian computation (ABC) provides evidence for a drastic decline in the effective population size approximately 0.3–0.5 million years ago (mya). We used simulations to show that this is unlikely to be a false-positive result due to the limited sample used here. To investigate the phylogenetic origin of Taiwan spruce, additional sequencing was performed in the Chinese spruce Picea wilsonii and combined with previously published data for three other mainland China species, Picea purpurea, Picea likiangensis, and P. schrenkiana. Analysis of population structure revealed that P. morrisonicola clusters most closely with P. wilsonii, and coalescent analyses using the program MIMAR dated the split to 4–8 mya, coincidental to the formation of Taiwan. Considering the population decrease that occurred after the split, however, led to a much more recent origin