Basis of Diagonally Alternating Harmonic Polynomials for low degree

Given a list of $n$ cells $L=[(p_1,q_1),...,(p_n, q_n)]$ where $p_i, q_i\in \textbf{Z}_{\ge 0}$, we let $\Delta_L=\det |{(p_j!)^{-1}(q_j!)^{-1} x^{p_j}_iy^{q_j}_i} |$. The space of diagonally alternating polynomials is spanned by $\{\Delta_L\}$ where $L$ varies among all lists with $n$ cells. For $a>0$, the operators $E_a=\sum_{i=1}^{n} y_i\partial_{x_i}^a$ act on diagonally alternating polynomials and Haiman has shown that the space $A_n$ of diagonally alternating harmonic polynomials is spanned by $\{E_\lambda\Delta_n\}$. For $t=(t_m,...,t_1)\in \textbf{Z}_{> 0}^m$ with $t_m>...>t_1>0$, we consider here the operator $F_t=\det\big\|E_{t_{m-j+1}+(j-i)}\big\|$. Our first result is to show that $F_t\Delta_L$ is a linear combination of $\Delta_{L'}$ where $L'$ is obtained by {\sl moving} $\ell(t)=m$ distinct cells from $L$ in some determined fashion. This allows us to control the leading term of some elements of the form $F_{t_{(1)}}... F_{t_{(r)}}\Delta_n$. We use this to describe explicit bases of some of the bihomogeneous components of $A_n=\bigoplus A_n^{k,l}$ where $A_n^{k,l}=\hbox{Span}\{E_\lambda\Delta_n :\ell(\lambda)=l, |\lambda|=k\}$. More precisely we give an explicit basis of $A_n^{k,l}$ whenever $k<n$. To this end, we introduce a new variation of Schensted insertion on a special class of tableaux. This produces a bijection between partitions and this new class of tableaux. The combinatorics of those tableaux $T$ allows us to know exactly the leading term of $F_T\Delta_n$ where $F_T$ is the operator corresponding to the columns of $T$ and whenever $n$ is bigger than the weight of $T$.Comment: To appear in JCT-A; 21 pages, one PDF figur

Multivariate Diagonal Coinvariant Spaces for Complex Reflection Groups

For finite complex reflexion groups, we consider the graded $W$-modules of diagonally harmonic polynomials in $r$ sets of variables, and show that associated Hilbert series may be described in a global manner, independent of the value of $r$.Comment: 12 pages, Removed a (wrong) conjecture, and reformulated in agreement. Also cleared up section on low degree term

Laminar-turbulent patterning in wall-bounded shear flows: a Galerkin model

On its way to turbulence, plane Couette flow - the flow between counter-translating parallel plates - displays a puzzling steady oblique laminar-turbulent pattern. We approach this problem via Galerkin modelling of the Navier-Stokes equations. The wall-normal dependence of the hydrodynamic field is treated by means of expansions on functional bases fitting the boundary conditions exactly. This yields a set of partial differential equations for the spatiotemporal dynamics in the plane of the flow. Truncating this set beyond lowest nontrivial order is numerically shown to produce the expected pattern, therefore improving over what was obtained at cruder effective wall-normal resolution. Perspectives opened by the approach are discussed.Comment: to appear in Fluid Dynamics Research; 14 pages, 5 figure

The Earth Mover\u27s Distance Through the Lens of Algebraic Combinatorics

The earth mover\u27s distance (EMD) is a metric for comparing two histograms, with burgeoning applications in image retrieval, computer vision, optimal transport, physics, cosmology, political science, epidemiology, and many other fields. In this thesis, however, we approach the EMD from three distinct viewpoints in algebraic combinatorics. First, by regarding the EMD as the symmetric difference of two Young diagrams, we use combinatorial arguments to answer statistical questions about histogram pairs. Second, we adopt as a natural model for the EMD a certain infinite-dimensional module, known as the first Wallach representation of the Lie algebra su(p,q), which arises in the Howe duality setting in Type A; in this setting, we show how the second fundamental theorem of invariant theory generalizes the northwest corner rule\u27\u27 from optimal transport theory, yielding a simple interpretation of the partial matching\u27\u27 case of the EMD via separation into invariants and harmonics. Third, we reapproach partial matching in the context of crystal bases of Types A, B, and C, which leads us to introduce a variation of the EMD in terms of distance on a crystal graph. Having exploited these three approaches, we generalize all of our EMD results to an arbitrary number of histograms rather than only two at a time. In the final chapter, we observe a combinatorial connection between generalized BGG resolutions arising in Type-A Howe duality and certain non-holomorphic discrete series representations of the group SU(p,q)