arXiv subjects
Publications and source records attributed to Tamar Friedmann.
We build on the methods introduced by Friedmann, Hanlon, Stanley, and Wachs, and further developed by Brauner and Friedmann, to construct additional classes of presentations of Specht modules. We obtain these presentations by defining a linear operator which is a symmetrized sum of dual Garnir relations on the space of column tabloids. Our presentations apply to the vast majority of shapes of Specht modules.
We continue our study, initiated in our prior work with Richard Stanley, of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra known as the free Filippov $n$-algebra with $k$ brackets. Our ultimate aim is to determine the multiplicities of the irreducible representations in this representation. This had been done for the ordinary Lie representation ($n=2$ case) by Kraskiewicz and Weyman. The $k=2$ case was handled in our prior work, where the representation was shown to be isomorphic to $S^{2^{n-1}1}$. In this paper, for general $n$ and $k$, we obtain decomposition results that enable us to determine the multiplicities in the $k=3$ and $k=4$ cases. In particular we prove that in the $k=3$ case, the representation is isomorphic to $S^{3^{n-1}1} \oplus S^{3^{n-2}21^2}$. Our main result shows that the multiplicities stabilize in a certain sense when $n$ exceeds $k$. As an important tool in proving this, we present two types of generalizations of the notion of Specht module that involve trees.
We obtain a new presentation for Specht modules whose conjugate shapes have strictly decreasing parts by introducing a linear operator on the space generated by column tabloids. The generators of the presentation are column tabloids and the relations form a proper subset of the Garnir relations of Fulton. The results in this paper extend earlier results of the authors and Stanley on Specht modules of staircase shape.
This paper continues the study of two numbers that are associated with Lie groups. The first number is $N(G,m)$, the number of conjugacy classes of elements in $G$ whose order divides $m$. The second number is $N(G,m,s)$, the number of conjugacy classes of elements in $G$ whose order divides $m$ and which have $s$ distinct eigenvalues, where we view $G$ as a matrix group in its smallest-degree faithful representation. We describe systematic algorithms for computing both numbers for $G$ a connected and simply-connected exceptional Lie group. We also provide explicit results for all of $N(G,m)$, $N(G_2,m,s)$, and $N(F_4,m,s)$. The numbers $N(G,m,s)$ were previously known only for the classical Lie groups; our results for $N(G,m)$ agree with those already in the literature but are obtained differently.
We generalize the derivation of the Wallis formula for $\pi$ from a variational computation of the spectrum of the Hydrogen atom. We obtain infinite product formulas for certain combinations of gamma functions, which include irrational numbers such as $\sqrt 2$ as well as some nested radicals. We also derive Euler's reflection formula for reciprocals of positive even integers. We show that Bohr's correspondence principle allows us to derive our product formulas and the reflection formula without the need for the limit definition of the gamma function.
Fulton and Kraskiewicz gave a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We simplify this presentation by showing that it can be generated by a single relation for each pair of columns of a tableau with ordered columns, thereby significantly reducing the number of generators given in the original construction. Our presentation applies to all Specht modules, and is of a similar nature to a recent result by Friedmann-Hanlon-Stanley-Wachs that applies to staircase partitions. We show that our presentation implies the Friedmann-Hanlon-Stanley-Wachs presentation.
We initiate a study of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group $S_{2n-1}$ on the multilinear component of the free LAnKe with $2n-1$ generators is given by an irreducible representation whose dimension is the $n$th Catalan number. This leads to a more general result on eigenspaces of a certain linear operator, which has additional consequences. We also obtain a new presentation of Specht modules of staircase shape as a consequence of our central result.
Low-cost coin vibrational motors, used in haptic feedback, exhibit rotational internal motion inside a rigid case. Because the motor case motion exhibits rotational symmetry, when placed into a fluid such as glycerin, the motor does not swim even though its vibrations induce steady streaming in the fluid. However, a piece of rubber foam stuck to the curved case and giving the motor neutral buoyancy also breaks the rotational symmetry allowing it to swim. We measured a 1 cm diameter coin vibrational motor swimming in glycerin at a speed of a body length in 3 seconds or at 3 mm/s. The swim speed puts the vibrational motor in a low Reynolds number regime similar to bacterial motility, but because of the vibration it is not analogous to biological organisms. Rather the swimming vibrational motor may inspire small inexpensive robotic swimmers that are robust as they contain no external moving parts. A time dependent Stokes equation planar sheet model suggests that the swim speed depends on a steady streaming velocity $V_{stream} \sim Re_s^{1/2} U_0$ where $U_0$ is the velocity of surface vibrations, and streaming Reynolds number $Re_s = U_0^2/(ων)$ for angular vibrational frequency $ω$ and fluid kinematic viscosity $ν$.
For $p$ an odd prime and $F$ the cyclic group of order $p$, we show that the number of conjugacy classes of embeddings of $F$ in $SU(p)$ such that no element of $F$ has 1 as an eigenvalue is $(1+C_{p-1})/p$, where $C_{p-1}$ is a Catalan number. We prove that the only coset space $SU(p)/F$ that admits a $p$-local $H$-structure is the classical Lie group $PSU(p)$. We also show that $SU(4)/\mathbb Z_3$, where $\mathbb Z_3$ is embedded off the center of $SU(4)$, is a novel example of an $H$-space, even globally. We apply our results to the study of homotopy classes of maps from $BF$ to $BSU(n)$.
A famous pre-Newtonian formula for $π$ is obtained directly from the variational approach to the spectrum of the hydrogen atom in spaces of arbitrary dimensions greater than one, including the physical three dimensions.
Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define $N(G,m)$ to be the number of conjugacy classes of elements of finite order $m$ in a Lie group $G$, and $N(G,m,s)$ to be the number of such classes whose elements have $s$ distinct eigenvalues or conjugate pairs of eigenvalues. What is $N(G,m)$ for $G$ a unitary, orthogonal, or symplectic group? What is $N(G,m,s)$ for these groups? For some cases, the first question was answered a few decades ago via group-theoretic techniques. It appears that the second question has not been asked before; here it is inspired by questions related to enumeration of vacua in string theory. Our combinatorial methods allow us to answer both questions.
We discuss the implications of our prior results obtained in our companion paper [arXiv:0910.2229]. Inescapably, they lead to three laws governing the size of hadrons, including in particular protons and neutrons that make up the bulk of ordinary matter: a) there are no radial excitations in low-energy QCD; b) the size of a hadron is largest in its ground state; c) the hadron's size shrinks when its orbital excitation increases. The second and third laws follow from the first law. It follows that the path from confinement to asymptotic freedom is a Regge trajectory. It also follows that the top quark is a free, albeit short-lived, quark. [For Note Added regarding experimental support, including the experiments studying muonic hydrogen, and other experiments, see last page.]
We propose a new schematic model for mesons in which the building blocks are quarks and flavor-antisymmetric diquarks. The outcome is a new classification of the entire meson spectrum into quark-antiquark and diquark-antidiquark states which does not give rise to a radial quantum number: all mesons which have so far been believed to be radially excited are orbitally excited diquark-antidiquark states; similarly, there are no radially excited baryons. Further, mesons that were previously viewed as "exotic" are no longer exotic as they are now naturally integrated into the classification as diquark-antidiquark states. The classification also leads to the introduction of isorons (iso-hadrons), which are analogs of atomic isotopes, and their magic quantum numbers, which are analogs of the magic numbers of the nuclear shell model. The magic quantum numbers of isorons match the quantum numbers expected for low-lying glueballs in lattice QCD. We observe that interquark forces in mesons behave substantially differently from those in baryons: qualitatively, they are color-magnetic in mesons but color-electrostatic in baryons. We comment on potential models and the hydrogen atom. The implications of our results for confinement, asymptotic freedom, and a new set of relations between two fundamental properties of hadrons - their size and their energy - are discussed in our companion paper [arXiv:0910.2231].
We derive formulas for counting certain classes of vacua in the string/M theory landscape. We do so in the context of the moduli space of M-theory compactifications on singular manifolds with G_2 holonomy. Particularly, we count the numbers of gauge theories with different gauge groups but equal numbers of U(1) factors which are dual to each other. The vacua correspond to various symmetry breaking patterns of grand unified theories. Counting these dual vacua is equivalent to counting the number of conjugacy classes of elements of finite order inside Lie groups. We also point out certain cases where the conventional expectation is that symmetry breaking patterns by Wilson lines and Higgs fields are the same, but we show they are in fact different.
We discover the unique, simple Lie Algebra of the Third Kind, or LATKe, that stems from codimension 6 orbifold singularities and gives rise to a kind of Yang-Mills theory which simultaneously is pure and contains matter. The root space of the LATKe is 1-dimensional and its Dynkin diagram consists of one point. The uniqueness of the LATKe is a vacuum selection mechanism.
The spectrum of the square of the angular momentum in arbitrary dimensions is derived using only group theoretical techniques. This is accomplished by application of the Lie algebra of the noncompact group O(2,1).
We construct a classical dynamical system whose phase space is a certain infinite-dimensional Grassmannian manifold, and propose that it is equivalent to the large N limit of two-dimensional QCD with an O(2N+1) gauge group. In this theory, we find that baryon number is a topological quantity that is conserved only modulo 2. We also relate this theory to the master field approach to matrix models.
We study Schwinger pair creation of charged Kaluza-Klein particles from a static KK electric field. We find that the gravitational backreaction of the electric field on the geometry - which is incorporated via the electric KK Melvin solution - prevents the electrostatic potential from overcoming the rest mass of the KK particles, thus impeding the tunneling mechanism which is often thought of as responsible for the pair creation. However, we find that pair creation still occurs with a finite rate formally similar to the classic Schwinger result, but via an apparently different mechanism, involving a combination of the Unruh effect and vacuum polarization due to the E-field.