# MATH 7382 • Representation theory of big groups and probability (Graduate topics course)

### Problem Sets

1. De Finetti’s Theorempdf | TeX Source | due: 10/2/12

2. Symmetric Functionspdf | TeX Source | due: 10/16/12

3. Total nonnegativity and determinantal processespdf | TeX Source | due: 11/27/12

4. Discrete Bessel kernel and discrete sine processpdf | TeX Source | due: 12/06/12

### Lectures

1. (9/7) Introduction | Slidespdf

CLT for concrete distributions. Random growth models (PNG, TASEP, push-block dynamics). Interlacing arrays and representations of finite unitary groups.

2. (9/11) De Finetti’s theorem: exchangeable binary sequences; harmonic functions; coherent systems on the Pascal triangle.

Convex spaces and their extreme points. Bernoulli (= independent exchangeable) sequences; harmonic functions $$\phi_p(a,b)=p^a(1-p)^b$$; extreme coherent systems (on the infinite Pascal triangle).

Extreme coherent systems on finite-$$N$$ subgraphs of the Pascal triangle. Approximation theorem.

3. (9/13) Boundary of a graph. Convergence of coherent systems. Approximation theorem.

Extreme coherent systems on finite-$$N$$ subgraphs of the Pascal triangle: explicit formula $$\dim(c,d)*\dim[(c,d);(a,b)]/\dim(a,b)$$. Limits of extreme coherent systems on finite-$$N$$ subgraphs. Proof of de Finetti’s theorem. Integral representation of coherent systems.

Linear functionals $$F$$ on $$\mathbb{R}[x,y]$$ with properties

• $$F(1)=1$$;
• $$F$$ vanishes on the ideal $$(x+y-1)\mathbb{R}[x,y]$$;
• $$F(x^ay^b)>=0$$.

Classification. Extreme functionals are multiplicative and vice versa.

4. (9/18) Final remarks on de Finetti’s theorem. Facts from representation theory of finite and compact groups: irreducible representations, characters, etc. Definition of characters of "big" groups: the infinite symmetric group and the infinite-dimensional unitary group. Symmetric polynomials. Schur polynomials.

5. (9/20) Schur polinomials as characters of the unitary groups. Orthogonality. Radial part of the Haar measure.

6. (9/25) Branching of Schur polynomials. Combinatorial formula for Schur polynomials. Semistandard Young tableaux. Dimensions of irreducible representations of the $$U(N)$$’s. Formulation of the Edrei-Voiculescu theorem about extreme characters of the infinite-dimensional unitary group.

7. (9/27) Discussion of the Edrei-Voiculescu theorem. Gelfand-Tsetlin graph. Start of proof of the Edrei-Voiculescu theorem - lemma that interlacing of particles can be encoded as a suitable determinant with virtual particles.

8. (10/02) Proof of the Edrei-Voiculescu theorem using the explicit formula for relative dimensions and the inverse Vandermonde matrix.

9. (10/04) End of the proof of the Edrei-Voiculescu theorem. Discussion. Random tilings.

New theme: algebra of symmetric functions.

10. (10/09) Algebra of symmetric functions. Definition. Linear bases. Multiplicative bases: $$e$$- and $$h$$-functions. Cauchy identity. Jacobi-Trudi identity. Pieri formula. Standard Young tableaux.

11. (10/11) Frobenius characteristic map. Branching of irreducible representations of symmetric groups. Extreme characters of the infintie symmetric group. Boundary of the Young graph: Edrei-Thoma theorem.

12. (10/16) Unified picture of extreme characters of infinite-dimensional unitary and infinite symmetric group. Connection to totally nonnegative sequences: one-sided (symmetric group) and two-sided (unitary group). Gessel-Viennot’s Lemma.

13. (10/18) Gessel-Viennot’s Lemma, its applications:

• Catalan-Hankel determinants;
• Schur polynomials and $$\mathrm{Dim}_N\lambda$$, proof of Jacobi-Trudi identity;
• Karlin-McGregor’s context of nonintersecting Markov dynamics.

Totally nonnegative (TN) Toeplitz matrices. TN sequences, TN probability measures on integers. Multiplication of Toeplitz matrices is the same as convolution of sequences, and the same as summation of independent random variables. Relation to graphs. Graphs-building blocks of TN sequences.

14. (10/24) Final remarks about Gessel-Viennot determinants and totally nonnegative Toeplitz matrices. Point processes. Definitions. Correlation functions. Determinantal point processes. Measures given by products of determinants.

15. (10/30) Point processes. Correlation measures and functions. Biorthogonal and orhtogonal ensembles. Examples. Projection correlation kernels. Characterization of determinantal processes with Hermitean symmetric kernels.

16. (11/1) Orthogonal polynomial ensembles. Correlation kernels with respect to the orthogonality and Lebesgue reference measures. Examples: CUE, GUE, gamma-case of the extreme coherent system for the infinite-dimensional unitary group. Connection of the latter measures to Schur-Weyl duality.

17. (11/6) Projection structure of the kernel operators for orthogonal polynomial ensembles. Connection with difference/differential operators for known orthogonal polynomials.

18. (11/8) Asymptotics of Plancherel measures via difference equations. Ulam’s problem. Limit shape of Plancherel-random Young diagrams.

19. (11/13) More limit shapes of random Young diagrams. Limit behavior of eigenvalues of GUE random matrices.

20. (11/15) Final remarks on the difference/differential operator method. Expressing correlation kernels as double contour integrals.

21. (11/27) Asymptotics of the poissonized Plancherel measure by saddle point analysis. Discrete sine process, Airy process. Tracy-Widom distribution, expression as a Fredholm determinant.

22. (11/29) Schur measures. Definition. Examples: poissonized Plancherel measure, Schur-Weyl measures, `rectangular’ measures, $$z$$-measures. Relation of $$z$$-measures to some distinguished reducible characters of the infinite symmetric group. Schur measures are determinantal point processes: proof using biorthogonal ensembles (Johansson). Application to our examples.

23. (12/4) Fock space approach (Okounkov) to the correlation kernel of the Schur measures. Skew Schur functions. Schur process.

24. (12/6) Application of the Schur process to random (skew) plane partitions (Okounkov-Reshetikhin 1, 2). Schur dynamics (Borodin). Macdonald symmetric functions: generalization of Schur functions. Idea of Macdonald processes and their dynamics (Borodin-Corwin). Some probabilistic applications.

### References

There is no single textbook. Lecture notes (in some form) will be posted, and relevant links to research papers (most of which are freely available at the arXiv) will be provided.

More material can be found in the following books:

• I.G. Macdonald. Symmetric functions and Hall polynomials. Oxford University Press, 1995.

• H. Weyl. The Classical Groups: Their Invariants and Representations. Princeton University Press, 1997.

• W. Fulton, J. Harris. Representation Theory: A First Course. Springer, 1991.