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Israel Gelfand, who made profound and prolific contributions to many areas of mathematics, including functional analysis, representation theory, operator algebras, and partial differential equations, died on Monday, age 96.

Gelfand’s beautiful theory of {C^*}-algebras and related spaces made quite an impact on me as a graduate student in Princeton, to the point where I was seriously considering working in this area; but there was not much activity in operator algebras at the time there, and I ended up working in harmonic analysis under Eli Stein instead. (Though I am currently involved in another operator algebras project, of which I hope to be able to discuss in the near future. The commutative version of Gelfand’s theory is discussed in these lecture notes of mine.)

I met Gelfand only once, in one of the famous “Gelfand seminars” at the IHES in 2000. The speaker was Tim Gowers, on his new proof of Szemerédi’s theorem. (Endre Szemerédi, incidentally, was Gelfand’s student.) Gelfand’s introduction to the seminar, on the subject of Banach spaces which both mathematicians contributed so greatly to, was approximately as long as Gowers’ talk itself!

There are far too many contributions to mathematics by Gelfand to name here, so I will only mention two. The first are the Gelfand-Tsetlin patterns, induced by an {n \times n} Hermitian matrix {A}. Such matrices have {n} real eigenvalues {\lambda_{n,1} \leq \ldots \leq \lambda_{n,n}}. If one takes the top {n-1 \times n-1} minor, this is another Hermitian matrix, whose {n-1} eigenvalues {\lambda_{n-1,1} \leq \ldots \leq \lambda_{n-1,n-1}} intersperse the {n} eigenvalues of the original matrix: {\lambda_{n,i} \leq \lambda_{n-1,i} \leq \lambda_{n,i+1}} for every {1 \leq i \leq n-1}. This interspersing can be easily seen from the minimax characterisation

\displaystyle  \lambda_{n,i} = \inf_{\hbox{dim}(V)=i} \sup_{v \in V: \|v\|=1} \langle Av, v \rangle

of the eigenvalues of {A}, with the eigenvalues of the {n-1 \times n-1} minor being similar but with {V} now restricted to a subspace of {{\mathbb C}^{n-1}} rather than {{\mathbb C}^n}.

Similarly, the eigenvalues {\lambda_{n-2,1} \leq \ldots \leq \lambda_{n-2,n-2}} of the top {n-2 \times n-2} minor of {A} intersperse those of the previous minor. Repeating this procedure one eventually gets a pyramid of real numbers of height and width {n}, with the numbers in each row interspersing the ones in the row below. Such a pattern is known as a Gelfand-Tsetlin pattern. The space of such patterns forms a convex cone, and (if one fixes the initial eigenvalues {\lambda_{n,1},\ldots,\lambda_{n,n}}) becomes a compact convex polytope. If one fixes the initial eigenvalues {\lambda_{n,1},\ldots,\lambda_{n,n}} of {A} but chooses the eigenvectors randomly (using the Haar measure of the unitary group), then the resulting Gelfand-Tsetlin pattern is uniformly distributed across this polytope; the {n=2} case of this observation is essentially the classic observation of Archimedes that the cross-sectional areas of a sphere and a circumscribing cylinder are the same. (Ultimately, the reason for this is that the Gelfand-Tsetlin pattern almost turns the space of all {A} with a fixed spectrum (i.e. the co-adjoint orbit associated to that spectrum) into a toric variety. More precisely, there exists a mostly diffeomorphic map from the co-adjoint orbit to a (singular) toric variety, and the Gelfand-Tsetlin pattern induces a complete set of action variables on that variety.) There is also a “quantum” (or more precisely, representation-theoretic) version of this observation, in which one can decompose any irreducible representation of the unitary group {U(n)} into a canonical basis (the Gelfand-Tsetlin basis), indexed by integer-valued Gelfand-Tsetlin patterns, by first decomposing this representation into irreducible representations of {U(n-1)}, then {U(n-2)}, and so forth.

The structure, symplectic geometry, and representation theory of Gelfand-Tsetlin patterns was enormously influential in my own work with Allen Knutson on honeycomb patterns, which control the sums of Hermitian matrices and also the structure constants of the tensor product operation for representations of {U(n)}; indeed, Gelfand-Tsetlin patterns arise as the degenerate limit of honeycombs in three different ways, and we in fact discovered honeycombs by trying to glue three Gelfand-Tsetlin patterns together. (See for instance our Notices article for more discussion. The honeycomb analogue of the representation-theoretic properties of these patterns was eventually established by Henriques and Kamnitzer, using {gl(n)} crystals and their Kashiwara bases.)

The second contribution of Gelfand I want to discuss is the Gelfand-Levitan-Marchenko equation for solving the one-dimensional inverse scattering problem: given the scattering data of an unknown potential function {V(x)}, recover {V}. This is already interesting in and of itself, but is also instrumental in solving integrable systems such as the Korteweg-de Vries equation, because the Lax pair formulation of such equations implies that they can be linearised (and solved explicitly) by applying the scattering and inverse scattering transforms associated with the Lax operator. I discuss the derivation of this equation below the fold.

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This problem lies in the highly interconnected interface between algebraic combinatorics (esp. the combinatorics of Young tableaux and related objects, including honeycombs and puzzles), algebraic geometry (particularly classical and quantum intersection theory and geometric invariant theory), linear algebra (additive and multiplicative, real and tropical), and the representation theory (classical, quantum, crystal, etc.) of classical groups. (Another open problem in this subject is to find a succinct and descriptive name for the field.) I myself haven’t actively worked in this area for several years, but I still find it a fascinating and beautiful subject. (With respect to the dichotomy between structure and randomness, this subject lies deep within the “structure” end of the spectrum.)

As mentioned above, the problems in this area can be approached from a variety of quite diverse perspectives, but here I will focus on the linear algebra perspective, which is perhaps the most accessible. About nine years ago, Allen Knutson and I introduced a combinatorial gadget, called a honeycomb, which among other things controlled the relationship between the eigenvalues of two arbitrary Hermitian matrices A, B, and the eigenvalues of their sum A+B; this was not the first such gadget that achieved this purpose, but it was a particularly convenient one for studying this problem, in particular it was used to resolve two conjectures in the subject, the saturation conjecture and the Horn conjecture. (These conjectures have since been proven by a variety of other methods.) There is a natural multiplicative version of these problems, which now relates the eigenvalues of two arbitrary unitary matrices U, V and the eigenvalues of their product UV; this led to the “quantum saturation” and “quantum Horn” conjectures, which were proven a couple years ago. However, the quantum analogue of a “honeycomb” remains a mystery; this is the main topic of the current post.

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