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Let $L(\lambda)$ be a highest weight Harish-Chandra module with highest weight $\lambda$. When the associated variety of $L(\lambda)$ is not maximal, that is, not equal to the nilradical of the corresponding parabolic subalgebra, we prove that the unitarity of $L(\lambda)$ can be determined by a simple condition on the value of $z = (\lambda + \rho, \beta^{\vee})$, where $\rho$ is half the sum of positive roots and $\beta$ is the highest root. In the proof, certain distinguished antichains of positive noncompact roots play a key role. By using these antichains, we are also able to provide a uniform formula for the Gelfand--Kirillov dimension of all highest weight Harish-Chandra modules, generalizing our previous result for the case of unitary highest weight Harish-Chandra modules., Comment: 20 pages

We introduce sets $\mathcal{R}(\sigma)$ consisting of semistandard tableaux of shape $\sigma$, subject to a certain restriction on the initial column. These $\mathcal{R}(\sigma)$ are generalizations of the rational, symplectic, or orthogonal tableaux (which furnish a weight basis for the irreducible finite-dimensional representations $U_\sigma$ of the classical groups $H = \operatorname{GL}_k$, $\operatorname{Sp}_{2k}$, or $\operatorname{O}_k$, respectively). As our main result, we show that the cardinality of $\mathcal{R}(\sigma)$ gives the Bernstein degree (i.e., multiplicity) of the modules of covariants (of type $U_\sigma$) of each classical group $H$. Via Howe duality, these modules can also be viewed as $(\mathfrak{g},K)$-modules of unitary highest weight representations of a real reductive group $G_{\mathbb{R}}$. Notably, our result using $\mathcal{R}(\sigma)$ is valid regardless of the rank of $H$, whereas the previous result of Nishiyama-Ochiai-Taniguchi (expressing the Bernstein degree in terms of $\dim U_\sigma$) holds only when $k$ is at most the real rank of $G_{\mathbb{R}}$. In effect, as $k$ increases beyond this range, $|\mathcal{R}(\sigma)|$ interpolates between $\dim U_\sigma$ and the dimension of a certain "limiting" $K$-module., Comment: 16 pages; certain parts of the exposition overlap with our paper arXiv:2312.16749, which contains further details

Let $n\in\mathbb{N}, n\geq 2$. We prove that the strongly singular differential operator \[\left(\Delta^2 +c|x|^{-4}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}, \quad c \in \mathbb{R}, \] is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if and only if \[c\geq \begin{cases}3(n+2)(6-n)&\mbox{for $2\leq n\leq 5$};\\[5pt] {\displaystyle -\frac{n(n+4)(n-4)(n-8)}{16}}&\mbox{for $n\geq 6$}.\end{cases}\] Via separation of variables, our proof reduces to studying the essential self-adjointness on the space $C_0^{\infty}((0,\infty))$ of fourth-order Euler-type differential operators of the form \[ \frac{d^4}{dr^4}+c_1\left(\frac{1}{r^2}\frac{d^2}{dr^2}+\frac{d^2}{dr^2}\frac{1}{r^2}\right)+\frac{c_2}{r^4},\quad r\in(0,\infty),\quad(c_1,c_2)\in \mathbb{R}^2,\] in $L^2((0,\infty);dr)$. Our methods generalize to differential operators related to higher-order powers of the Laplacian, however, there are some nontrivial subtleties that arise. For example, the natural expectation that for $m,n\in\mathbb{N}$, $n \geq 2$, there exist $c_{m,n}\in\mathbb{R}$ such that $\left(\Delta^m+c|x|^{-2m}\right)\big|_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}$ is essentially self-adjoint in $L^2(\mathbb{R}^n; d^n x)$ if and only if $c \geq c_{m,n}$, turns out to be false. Indeed, for $n=20$, we prove that the differential operator \[ \left((-\Delta)^5+c|x|^{-10}\right)\big|_{C_0^{\infty}(\mathbb{R}^{20} \backslash \{0\})}, \quad c \in \mathbb{R},\] is essentially self-adjoint in $L^2\big( \mathbb{R}^{20}; d^{20} x\big)$ if and only if $c\in [0,\beta]\cup [\gamma,\infty)$, where $\beta\approx 1.0436\times 10^{10}$, and $\gamma\approx 1.8324\times 10^{10}$ are the two real roots of the quartic equation \begin{align*}&3125z^4-83914629120000z^3+429438995162964368031744 z^2\\&\quad+1045471534388841527438982355353600z\\&\quad +629847004905001626921946285352115240960000=0.\end{align*}, Comment: 24 pages

We prove a simple formula that calculates the associated variety of a highest weight Harish-Chandra module directly from its highest weight. We also give a formula for the Gelfand--Kirillov dimension of highest weight Harish-Chandra module which is uniform across Cartan types and is valid for arbitrary infinitesimal character., Comment: 34 pages

We reconsider an old problem, namely the dimension of the $G$-invariant subspace in $V^{\otimes p} \otimes V^{*\otimes q}$, where $G$ is one of the classical groups ${\rm GL}(V)$, ${\rm SL}(V)$, ${\rm O}(V)$, ${\rm SO}(V)$, or ${\rm Sp}(V)$. Spanning sets for the invariant subspace have long been well known, but linear bases are more delicate. Beginning in the broader setting of polynomial invariants, we write down multigraded linear bases which we realize as certain arc diagrams (which may include hyperedges); then the bases for tensor invariants are obtained by restricting our attention to the diagrams that are 1-regular. We survey the many equivalent ways -- some old, some new -- to enumerate these diagrams., Comment: 16 pages + appendix

For a complex reductive group $H$ with finite-dimensional representations $W$ and $U$, the module of covariants for $W$ of type $U$ is the space of all $H$-equivariant polynomial functions $W \longrightarrow U$. In this paper, we take $H$ to be one of the classical groups $\operatorname{GL}(V)$, $\operatorname{Sp}(V)$, or $\operatorname{O}(V)$ arising in Howe's dual pair setting, where $W$ is a direct sum of copies of $V$ and $V^*$. Our main result is a uniform combinatorial model for Stanley decompositions of the modules of covariants, using visualizations that we call jellyfish. Our decompositions allow us to interpret the Hilbert series as a positive combination of rational expressions which have concrete combinatorial interpretations in terms of lattice paths; significantly, this interpretation does not depend on the Cohen-Macaulay property. As a corollary, we recover a major result of Nishiyama-Ochiai-Taniguchi (2001) regarding the Bernstein degree of unitary highest weight $(\mathfrak{g},K)$-modules. We also extend our methods to compute the Hilbert series of the invariant rings for the groups $\operatorname{SL}(V)$ and $\operatorname{SO}(V)$, as well as the Wallach representations of type ADE., Comment: 46 pages; this article draws heavily from, and largely supersedes, the second half of our previous article arXiv:2310.04581. The main result in the present article generalizes and uniformizes the many examples in that previous article. Version 3 corrects typos and improves the exposition

We consider the generalized eigenvalue problem for the classical Euler differential equation and demonstrate its intimate connection with Meijer's $G$-functions. In the course of deriving the solution of the generalized Euler eigenvalue equation we review some of the basics of generalized hypergeometric functions and Meijer's $G$-functions and some of its special cases where the underlying Mellin-type integrand exhibits higher-order poles., Comment: 21 pages

We consider essential self-adjointness on the space $C_0^{\infty}((0,\infty))$ of even order, strongly singular, homogeneous differential operators associated with differential expressions of the type \[ \tau_{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x > 0, \; n \in \mathbb{N}, \; c \in \mathbb{R}, \] in $L^2((0,\infty);dx)$. While the special case $n=1$ is classical and it is well-known that $\tau_2(c)\big|_{C_0^\infty((0,\infty))}$ is essentially self-adjoint if and only if $c \geq 3/4$, the case $n \in \mathbb{N}$, $n \geq 2$, is far from obvious. In particular, it is not at all clear from the outset that \[ \text{ there exists } c_n \in \mathbb{R}, \, n \in \mathbb{N}, \text{ such that } \tau_{2n}(c)\big|_{C_0^\infty((0,\infty))} \, \text{ is essentially self-adjoint if and only if } c \geq c_n. \tag{*}\label{0.1} \] As one of the principal results of this paper we indeed establish the existence of $c_n$, satisfying $c_n \geq (4n-1)!!\big/2^{2n}$, such that property \eqref{0.1} holds. In sharp contrast to the analogous lower semiboundedness question, \[ \text{ for which values of } c \, \text{\it is } \tau_{2n}(c)\big|_{C_0^{\infty}((0,\infty))} \, \text{ bounded from below?}, \] which permits the sharp (and explicit) answer $c \geq [(2n -1)!!]^{2}\big/2^{2n}$, $n \in \mathbb{N}$, the answer for \eqref{0.1} is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, \[ c_1 = 3/4, \quad c_2= 45, \quad c_3 = 2240 \big(214+7 \sqrt{1009}\,\big)\big/27, \] and remark that $c_n$ is the root of a polynomial of degree $n-1$. We demonstrate that for $n=6,7$, $c_n$ are algebraic numbers not expressible as radicals over $\mathbb{Q}$ (and conjecture this is in fact true for general $n \geq 6$)., Comment: 36 pages

The first half of this paper is largely expository, wherein we present a systematic combinatorial approach to the theory of polynomial (semi)invariants and multilinear invariants of several vectors and covectors, for the classical groups. This culminates in a graphical description of graded linear bases. By applying well-known results of lattice path combinatorics to Weyl's fundamental theorems of classical invariant theory, we write down Stanley decompositions and Hilbert-Poincare series in terms of families of non-intersecting lattice paths, enumerated with respect to certain corners. In the second half of the paper, we revisit the (semi)invariants in the first half as a special case of a much broader phenomenon. On one hand, polynomial invariants of a group $H$ can be generalized to modules of covariants, i.e., $H$-equivariant polynomial functions between $H$-modules. On the other hand, from the perspective of Roger Howe's theory of dual pairs, these modules of covariants can be viewed as infinite-dimensional simple $(\mathfrak{g}, K)$-modules. This suggests an expanded program in which our goal is to apply combinatorial techniques involving lattice paths in order to write down Hilbert series for arbitrary unitarizable highest-weight $(\mathfrak{g},K)$-modules. As a preview of future work in this program, we present examples showing how modules of covariants -- even those which are not Cohen-Macaulay, and therefore which we would not expect to be combinatorially nice -- can be decomposed in terms of lattice paths. We also extend these methods beyond the classical groups., Comment: 59 pages

An almost self-conjugate (ASC) partition has a Young diagram in which each arm along the diagonal is exactly one box longer than its corresponding leg. Classically, the ASC partitions and their conjugates appear in two of Littlewood's symmetric function identities. These identities can be viewed as Euler characteristics of BGG complexes of the trivial representation, for classical Hermitian symmetric pairs. In this paper, we consider partitions in which the arm-leg difference is an arbitrary constant $m$. Regarding these partitions and their conjugates as highest weights, we prove an identity yielding an infinite family of dimension equalities between $\mathfrak{gl}_n$- and $\mathfrak{gl}_{n+m}$-modules. We then interpret this combinatorial result in the context of blocks in parabolic category $\mathcal{O}$: using Enright-Shelton reduction, we find six infinite families of congruent blocks whose corresponding posets of highest weights consist of the partitions in question. These posets, in turn, lead to generalizations of the Littlewood identities and their corresponding BGG complexes. Our results in this paper shed light on the surprising combinatorics underlying the work of Enright and Willenbring (2004)., Comment: 33 pages, 5 figures, 4 tables

Given a natural number $n\geq3$ and two points $a$ and $b$ in the unit disk $\mathbb D$ in the complex plane, it is known that there exists a unique elliptical disk having $a$ and $b$ as foci that can also be realized as the intersection of a collection of convex cyclic $n$-gons whose vertices fill the whole unit circle $\mathbb T$. What is less clear is how to find a convenient formula or expression for such an elliptical disk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful tool for finding such a formula for some values of $n$. The main idea is to realize the elliptical disk as the numerical range of a matrix and the problem reduces to finding the eigenvalues of that matrix., Comment: 19 pages, 9 figures

We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szeg\H{o} polynomials. Our results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions. This paper contains new results, some old results presented from a different perspective or with a different proof, and a formal foundation for our analysis. We give a rigorous definition of the Poncelet property, of curves tangent to a family of polygons, and of polygons associated with Poncelet curves. As a result, we are able to clarify some misconceptions that appear in the literature and present counterexamples to some existing assertions along with necessary modifications to their hypotheses to validate them. For instance, we show that curves inscribed in some families of polygons supported on T are not necessarily convex, can have cusps, and can even intersect the unit circle. Two ideas play a unifying role in this work. The first is the utility of OPUC and the second is the advantage of working with tangent coordinates. This latter idea has been previously exploited in the works of B. Mirman, whose contribution we have tried to put in perspective., Comment: 41 pages. 11 figures

By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called Pieri inclusions and were first studied by Weyman in his thesis and described explicitly by Olver. More recently, these maps have appeared in the work of Eisenbud, Fl{\o}stad, and Weyman and of Sam and Weyman to compute pure free resolutions for classical groups. In this paper, we give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.

We give a uniform realization of the minimal representation of a double cover of the conformal group SO(2,n+1)_0 in the kernel of the wave operator on flat Minkowski space as a positive energy representation H^+ for n even and odd. Using this realization, we obtain an explicit orthonormal basis for H^+ that is well behaved with respect to energy and angular momentum. Of special note, for n odd, all functions in our basis are rational functions. Finally, using Fourier analysis with respect to this basis, we prove that every classical real-valued solution to the wave equation is the real part of a unique continuous element in the representation H^+., Comment: 13 pages, 1 figure

We show that the Lusternik-Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of G., Comment: 10 pages, 2 figures

Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity free. If V_{\lambda} is an irreducible representation of K, let S_{\lambda} denote the corresponding isotypic component of S. Write S_{\lambda} S_{\mu} for the subspace of S spanned by products of S_{\lambda} and S_{\mu}. If V_{\nu} occurs as an irreducible constituent of the tensor product of V_{\lambda} and V_{\mu}, is it true that S_{\nu} is contained in S_{\lambda} S_{\mu}? We investigate this question for representations arising in the context of Hermitian symmetric pairs. We show that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. We also show how the conjecture connects multiplication in the ring S to the usual Littlewood-Richardson rule.

In this paper the authors investigate infinite-dimensional representations $L$ in blocks of the relative (parabolic) category ${\mathcal O}_S$ for a complex simple Lie algebra, having the property that the cohomology of the nilradical with coefficients in $L$ ``looks like'' the cohomology with coefficients in a finite-dimensional module, as in Kostant's theorem. A complete classification of these ``Kostant modules'' in regular blocks for maximal parabolics in the simply laced types is given. A complete classification is also given in arbitrary (singular) blocks for Hermitian symmetric categories., Comment: 25 pages, 2 tables, 6 figures, uses PSTricks; v.2: added Secs. 4 (BGG resolutions) and 7.5 (Minimal free resolutions), expanded Introduction

Concept learning provides a natural framework in which to place the problems solved by the quantum algorithms of Bernstein-Vazirani and Grover. By combining the tools used in these algorithms--quantum fast transforms and amplitude amplification--with a novel (in this context) tool--a solution method for geometrical optimization problems--we derive a general technique for quantum concept learning. We name this technique "Amplified Impatient Learning" and apply it to construct quantum algorithms solving two new problems: BATTLESHIP and MAJORITY, more efficiently than is possible classically., Comment: 20 pages, plain TeX with amssym.tex, related work at http://www.math.uga.edu/~hunziker/ and http://math.ucsd.edu/~dmeyer/