Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials and may be viewed as K-theoretic analogs of key polynomials. The latter two polynomials have combinatorial formulas involving tableaux: Lascoux and Schützenberger gave a combinatorial formula for key polynomials using right keys; Buch gave a set-valued tableau formula for Grassmannian stable Grothendieck polynomials. We establish a novel combinatorial description for Lascoux polynomials involving right keys and set-valued tableaux. Our description generalizes the tableaux formulas of key polynomials and Grassmannian stable Grothendieck polynomials. To prove our description, we construct a new abstract Kashiwara crystal structure on set-valued tableaux. This construction answers an open problem of Monical, Pechenik and Scrimshaw.
Mathematics Subject Classifications: 05E05
Keywords: Lascoux polynomials, set-valued tableaux, crystal operators
The aim of this thesis proposed here is to design a novel, universal route of constructing a solid-phase catalytic system, which presents an acid-base multifunctional property. Such a catalytic system is capable of selectively promoting multiple one-pot tandem reactions(e.g. Deacetalization-Henry reaction, Cyano-Ethoxycarbonylation reaction……). The heterogeneous systems we utilized to perform tandem reactions would be chemically incompatible if carried out in solution or in multiple separate stages. A commercial available mesoporous silica, SBA-15, was used as support in this thesis and the spatial separation of two functional groups was achieved using known grafting chemistry followed by Ultra-violet/Ozonolysis treatment(Uv/O3). In the second chapter, a combination of simplest base, amino group(NH2-), and sulfonic acid(HSO3-) group was grafted onto commercial mesoporous SBA-15. Resulting catalyst was investigated by multiple characterization methods such as solid state 13C, 29Si cross-polarization magic-angle spinning (CP/MAS) NMR, Fourier-transform infrared spectrometer and N2 adsorption−desorption isotherms. Those data indicated both functional groups were successfully tethered onto SBA15 and original mesoporous structure was maintained well. Monitored by Gas-Chromatograph, the catalytic performance for a series of NH2-HSO3-SBA15 catalyst was examined in the second Chapter via Deacetalization-Henry reaction. Phyisical mixture and mono-functionalized catalysts(NH2 grafted SBA-15 and HSO3 grafted SBA-15) were tested as comparison groups as well. The result further confirmed the successfully grafting work of combining and separating both acid and base onto one support. The third Chapter deals with a new acid-base system, a phosphonic acid-cinchonidine bifunctional catalyst. Besides all techniques previous applied, solid state 31P cross-polarization magic-angle spinning (CP/MAS) NMR and CNS Flash elemental analysis were further introduced to characterize those more complicated parts. A bimodal-shaped pore sizes also revealed that Uv/ozonolysis process was slow but tunable by controlling the treatment length. Due to new functionalities introduce by grafting cinchonidine group, new system was further capable for a couple of new reactions. Three reactions were then successfully tested: Deacetalization-Henry cascade reaction, Cyano-Ethoxycarbonylation reaction and 1,4-addition of thiols to cylic enones. The result confirmed the successfully construction work of grafting new acid and base groups. However, a loss in enantioselectivity have been observed here. It may cause by non-chirality of tethered acid sites.
Lascoux polynomials simultaneously generalize two famous families of polynomialsarising from geometry and representation theory: They are non-symmetric analogs of Grassmannian stable Grothendieck polynomials, which represent Schubert classes in the connective K-theory of Grassmannians. Additionally, they serve as non- homogeneous analogs of key polynomials, the characters of Demazure modules. Both of these families have classical combinatorial formulas involving tableaux. We further generalize several of these formulas by establishing two combinatorial formulas for Lascoux polynomials.
We introduce a new row insertion algorithm on decreasing tableaux and increasing tableaux, generalizing Edelman-Greene (EG) row insertion. Our row insertion algorithm is a nontrivial variation of Hecke column insertion which generalizes EG column insertion. Similar to Hecke column insertion, our row insertion is bijective and respects Hecke equivalence, and therefore recovers the expansions of stable Grothendieck functions into Grassmannian stable Grothendieck functions.
Keywords: Hecke insertion, Grothendieck polynomials
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