Table of Contents
Advanced Discrete Mathematics
The following are notes that I have taken while studying the course MAST90030: Advanced Discrete Mathematics by Arun Ram.
(30/07/2025) Lecture 1:Handwritten Notes
Key words: Young's lattice, Brauer Bratteli diagram, Temperley-Lieb Bratteli diagram, Pascal's triangle, factorial, double factorial, Catalan, binomial coefficient, Catalan numbers.
(30/07/2025) Problem Solving/Questions 1:Handwritten Notes
Key words: source of Bratteli diagrams, correspondences, Brauer algebra, Temperley-Lieb algebra.
(18/07/2025) PS1 Exercises 1-3:Handwritten Notes
Key words: partitions, coarm - coleg, Jucys-Murphy elements, coleg - 1, Springer fibers.
(19/07/2025) PS1 Exercises 4-5:Handwritten Notes
Key words: Grassmanian, standard tableau, hook length formula.
(31/07/2025) Lecture 2:Handwritten Notes
Key words: n choose k, C[[x]], C((x)), C[x], exponential, binomial coefficient.
(31/07/2025) Problem Solving/Questions 2:Handwritten Notes
Key words: Representations of algebras, proof by induction, evaluating (x+y)^n, where does multiplication come from?.
(25/07/2025) PS2 Exercise 1 (WIP 1):Handwritten Notes
Key words: counting k-subsets of an n-set.
(25/07/2025) PS2 Exercise 1:Handwritten Notes
Key words: counting k-subsets of an n-set.
(01/08/2025) Lecture 3:Handwritten Notes
Key words: writing advice, symmetric group, permutation.
(06/08/2025) Lecture 4:Handwritten Notes
Key words: partition, box, length, weight, conjugate, standard tableau, f_lambda, hook-length formula, (co)arm, (co)leg.
(06/08/2025) Problem Solving/Questions 4:Handwritten Notes
Key words: functions, sum of squares of a row of the Brauer Bratteli diagram, identifying category-theoretic language in writing.
(07/08/2025) Lecture 5:Handwritten Notes
Key words: transposition, reflection, simple reflection, symmetric group, general linear group, k-cycle, Coxeter element, conjugacy class.
(07/08/2025) Problem Solving/Questions 5:Handwritten Notes
Key words: referencing, tenants of discrete mathematics, writing advice, G-set, G-action, orbit, stabiliser, orbit-stabiliser, loop Grassmanian, Grassmanian.
(08/08/2025) Lecture 6:Handwritten Notes
Key words: subset lattice, Young lattice, relation, poset, Hasse diagram, maximal chain, infimum, supremum, lattice, modular lattice.
(13/08/2025) Lecture 7:Handwritten Notes
Key words: writing advice, ranked modular lattice, q-analogues of integers and binomial coefficient, Fano plane.
(13/08/2025) Problem Solving/Questions 7:Handwritten Notes
Key words: quantum groups at q=0 and q=1, topologies and metrics on posets.
(14/08/2025) Lecture 8:Handwritten Notes
Key words: morphism of posets, isomorphism of posets, automorphism of posets, automorphism group, flag variety, standard flag.
(14/08/2025) Problem Solving/Questions 8:Handwritten Notes
Key words: Coxeter system, Coxeter group, Cayley graphs, Fano plane example.
(15/08/2025) Lecture 9:Handwritten Notes
Key words: projective space, Grassmanian, flag variety, maximal standard parabolic subspaces, Borel subgroup, Bruhat decomposition.
(20/08/2025) Lecture 10:Handwritten Notes
Key words: proof of counting theorem (i.e. Card(GLn(Fq)), Card(B), Card(Pk), etc.).
(21/08/2025) Lecture 11:Handwritten Notes
Key words: Hecke algebra, representation of Hecke algebra, Coxeter presentation of Sn.
(22/08/2025) Lecture 12:Handwritten Notes
Key words: proof of presentation theorems for Sn and GLn(Fq).
(27/08/2025) Lecture 13:Handwritten Notes
Key words: symmetric functions, Laurent polynomials, power sum, complete/homogeneous, elementary, monomial, gr, q-Pochammer, gr tilde, qr, one-row symmetric functions.
(28/08/2025) Lecture 14:Handwritten Notes
Key words: explicit formula for qr, monomial symmetric functions.
(28/08/2025) Problem Solving/Questions 14:Handwritten Notes
Key words: what makes vector spaces special?, Hall inner product.
(29/08/2025) Lecture 15:Handwritten Notes
Key words: monomial symmetric functions, q-binomial theorem, explicit formula for gr, Cauchy-Macdonald identity, Macdonald inner product.
(29/08/2025) Problem Solving/Questions 15:Handwritten Notes
Key words: how do you prove Bruhat decomposition?.
(03/09/2025) Lecture 16:Handwritten Notes
Key words: Cauchy-Macdonald kernel.
(03/09/2025) Problem Solving/Questions 16:Handwritten Notes
Key words: er and characteristic polynomial of a matrix, pr and matrix powers, Newton's identities, Wronski identities, total positivity.
(04/09/2025) Lecture 17:Handwritten Notes
Key words: binomial theorem, q-binomial theorem, prinicipal specialisation.
(04/09/2025) Problem Solving/Questions 17:Handwritten Notes
Key words: generalised cohomology theories, quantum cohomology.
(05/09/2025) Lecture 18:Handwritten Notes
Key words: crystals, WB-crystals, roots and simple roots, framework, i-strings, tensor product of crystals.
(10/09/2025) Lecture 19:Handwritten Notes
Key words: morphism of crystals, subcrystals, irreducible crystals, highest weight elements, character of a crystral, connected components.
(11/09/2025) Lecture 20:Handwritten Notes
Key words: words, decomposing crystals into irreducible components, tableaux, Arabic reading word, RSK correspondence.
(11/09/2025) Problem Solving/Questions 20:Handwritten Notes
Key words: invariant theory, GIT quotient.
(12/09/2025) Lecture 21:Handwritten Notes
Key words: Young lattice and decomposition rules, RSK correspondence/Schur-Weyl duality, defining Kashiwara operators on arbitrary elements, Littelmann paths.
(17/09/2025) Lecture 22:Handwritten Notes
Key words: Schur functions, alternating polynomials, Vandermonde determinant, Weyl denominator formula, Boson-Fermion correspondence.
(18/09/2025) Lecture 23:Handwritten Notes
Key words: characters of crystals are Schur functions, Weyl character formula.
(18/09/2025) Problem Solving/Questions 23:Handwritten Notes
Key words: shuffle algebra, Schur polynomials are irreducible characters of polynomials representations of GLn(C).
(19/09/2025) Lecture 24:Handwritten Notes
Key words: lattice permutation, skew shapes, skew tableaux, Littlewood-Richardson filling, skew Schur function, Littlewood-Richardson coefficients.
(24/09/2025) Lecture 25:Handwritten Notes
Key words: RSK, Cauchy identity, weight of a matrix, insertion.
(25/09/2025) Lecture 26:Handwritten Notes
Key words: RSK, Schur functions, specialisations of the Cauchy identity.
(25/09/2025) Problem Solving/Questions 26:Handwritten Notes
Key words: Worked examples of RSK.
(08/10/2025) Lecture 28:Handwritten Notes
Key words: Pieri rules, border strip, UDr, vertical strip.
(09/10/2025) Lecture 29:Handwritten Notes
Key words: simple Sn modules, Pieri rules, border strip, vertical strip, horizontal strip, Murnaghan-Nakayama rule, character tables, contents of boxes.
(10/10/2025) Lecture 30:Handwritten Notes
Key words: representations of Sk and Hk(q), C-algebra, module, simple module, irreducible module, group algebra, CSk, Jucys-Murphys elements, classifying simple Sk modules, Hecke algebra Hk(q).
(15/10/2025) Lecture 31:Handwritten Notes
Key words: group algebra of symmetric group, finite Hecke algebra, affine Hecke algebra, affine group, affine Lie algebra, classifying simple Hk(q)-modules.
(16/10/2025) Lecture 32:Handwritten Notes
Key words: double affine Hecke algebra (DAHA), referee report writing advice, T gamma mu, relating characters and symmetric functions, Schur-Weyl duality, Frobenius character formula.
(17/10/2025) Lecture 33:Handwritten Notes
Key words: Grassmannian, projective space, flag variety, presentation of Gln(Fq), building relations, greedy reduced word, Bruhat decomposition.
(22/10/2025) Lecture 34:Handwritten Notes
Key words: binomial theorem, q-analogue of power function, hypergeometric functions, q-hypergeometric function, q-analogue of binomial theorem, difference equations.
(23/10/2025) Lecture 35:Handwritten Notes
Key words: gamma function, q-gamma function, beta function, q-beta function, Gauss hypergeometric function, q-hypergeometric function, exponential function, q-exponential function.
(24/10/2025) Lecture 36:Handwritten Notes
Key words: q-integral / Jackson integral, q-differentiation, product rule, how to prepare a talk.
Exactly Solvable Models
The following are notes that I have taken while studying the course MAST90065: Exactly Solvable Models by Michael Wheeler.
(28/07/2025) Lecture 1:Handwritten Notes
Key words: Heisenberg magnet, Pauli matrices, su_2(C), spin states, Hamiltonian for Heisenberg magnet, XXZ Heisenberg spin chain, transfer matrix.
(28/07/2025) Exercise 1.3-1.4:Handwritten Notes
Key words: Explicitly computing Basis(V) and Hamiltonian for N=3.
(29/07/2025) Lecture 2:Handwritten Notes
Key words: transfer matrix, Lax matrix, conserved quantities, permutation matrix.
(29/07/2025) Exercise 1.5-1.6:Handwritten Notes
Key words: Rewriting L_k(u), explicitly computing T(u) for N=2.
(29/07/2025) Lecture 3:Handwritten Notes
Key words: properties of the permutation matrix, u=gamma/2 specialisation of monodromy and transfer matrix, deriving formula for Hamiltonian, transfer matrix commutes with Hamiltonian.
(04/08/2025) Lecture 3 Details:Handwritten Notes
Key words: (see Lecture 3).
(05/08/2025) Lecture 5:Handwritten Notes
Key words: RLL equation, R matrix, R check, RTT equation, unitarity equation, proof of commuting transfer matrices.
(08/08/2025) Lecture 6:Handwritten Notes
Key words: Algebraic Bethe Ansatz, Yang-Baxter Algebra, pseudo-vacuum.
(11/08/2025) Lecture 7:Handwritten Notes
Key words: creation operator, f(u,v), g(u,v), A(u) and D(u) acting on {v_1,...,v_n}.
(12/08/2025) Lecture 8:Handwritten Notes
Key words: Bethe vectors, Bethe equations, further prospects in the study of the Heisenberg spin chain.
(15/08/2025) Lecture 9:Handwritten Notes
Key words: Spin subspaces.
(18/08/2025) Lecture 10-12:Handwritten Notes
Key words: six-vertex model, vertex, state, ice rule, weight, partition function, internal/external edge, free energy, weights, Yang-Baxter equation, Unitarity relation, Yang-Baxter calculus, permutation graph.
(26/08/2025) Lecture 13-15 (Part 1):Handwritten Notes
Key words: relating QISM and six vertex model, A(u), B(u), C(u), D(u), t(u) and so on as partition functions, commutativity of B-matrices, components of Bethe vectors, domain wall partition function, Korepin's conditions.
(27/08/2025) Lecture 13-15 (Part 2):Handwritten Notes
Key words: proof of Korepin's conditions, Korepin's conditions uniquely define domain wall partition function, Izergin's determinant.
(01/09/2025) Lecture 16:Handwritten Notes
Key words: integrable probability, stochastic R-matrix, desymmetrise b-weights, graphical gauge transformation, invariant of configurations.
(02/09/2025) Lecture 17:Handwritten Notes
Key words: proof that b+-weight - b-weight is inavariant of configurations, defining stochastic six-vertex model, stochastic six-vertex model in a quadrant.
(05/09/2025) Lecture 18:Handwritten Notes
Key words: distribution functions, height function, q-moment of height function.
(08/09/2025) Lecture 19:Handwritten Notes
Key words: the function G S/T, spectral decomposition of a matrix, decomposition of the identity, eigenvectors of G S/T, F S, Cauchy identity for G S/T.
(09/09/2025) Lecture 20:Handwritten Notes
Key words: Proof of the Cauchy identity for G S/T.
(15/09/2025) Lecture 22:Handwritten Notes
Key words: F S orthogonality, integral formula/spectral decomposition for G S/T.
(16/09/2025) Lecture 23:Handwritten Notes
Key words: ASEP, analogues of G S/T and F S on the full integer line, translation invariance.
(19/09/2025) Lecture 24:Handwritten Notes
Key words: integral formula for S/T on the full integer line, limit of stochastic six-vertex model to ASEP, formula for transition probabilities of ASEP.
(22/09/2025) Lecture 25:Handwritten Notes
Key words: algebraic combinatorics, Schur polynomials, stability property, Jacobi-Trudi formula, branching rule, interlacing partitions.
(23/09/2025) Lecture 26:Handwritten Notes
Key words: equivalence of tableaux and row decomposition formula for Schur polynomials, Cauchy identitiy, dual Cauchy identity, Pieri identities, another six-vertex model.
(06/10/2025) Lecture 28:Handwritten Notes
Key words: Functions G S/T and F S for another six-vertex model, Cauchy identity.
(07/10/2025) Lecture 29:Handwritten Notes
Key words: reductions of the skew Cauchy identity, partition map, LGV, Jacobi-Trudi formula.
(10/10/2025) Lecture 30:Handwritten Notes
Key words: Jacobi-Trudi formula, hidden symmetry in G S/T and F S, concatenation lemma.
(14/10/2025) Lecture 31:Handwritten Notes
Key words: Schur measure, ascending Schur measure, plane partition measure, plane partitions, principal specialisation, markov dynamics relating Schur measures.
(17/10/2025) Lecture 32:Handwritten Notes
Key words: GUE corners process, Plancherel specialisation, power sum symmetric functions, Gaussian Unitary Ensemble (GUE).
(20/10/2025) Lecture 34:Handwritten Notes
Key words: ascending Schur measure, joint probability distribution of GUE corners process, integral formula for schur polynomial, plancherel specialisation of schur polynomial, saddle point analysis.
(25/10/2025) Lecture 35:Handwritten Notes
Key words: fermions and Clifford algebra, Fock representation, charge, partition basis, fermionic representation of a partition.
(27/10/2025) Lecture 36:Handwritten Notes
Key words: dual vacuum, expectation values, Wick's theorem, fermionic fields, Cauchy determinant, exam notes.