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1. Eigenvalues of Hermitian Matrices and Schubert Calculus

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Eigenvalues of Hermitian Matrices and Schubert Calculus AROUND THE HORN CONJECTURE L MANIVEL Abstract We discuss the problem of determining the p ossible sp ectra of a sum of Hermitian matrices each with known sp ectrum We explain how the Horn conjecture which gives a complete answer to this question is related with algebraic geometry symplectic geometry and representation theory The rst lecture is an introduction to Schubert calculus from which one direction of Horns conjecture can b e deduced The reverse direction follows from an application of geometric invariant theory this is treated in the second lecture Finally we explain in the third lecture how a version of Horns problem for sp ecial unitary matrices is related to the quantum cohomology of Grassmannians Notes for Gael VI I I CIRM March Eigenvalues of hermitian matrices and Schubert calculus The problem Let A B C b e complex Hermitian n by n matrices Denote the set of eigenvalues or sp ectrum of A by A A A and similarly by B and C the sp ectra n of B and C The main theme of these notes is the following question Suppose that A B C What can be their spectra A B C There are obvious relations like trace C trace A trace B or C A B But a complete answer to this longstanding question was given only recently and combines works and ideas from representation theory symplectic and algebraic geometry Weyls inequalities There are various characterizations of the eigenvalues of Hermitian ma trices many of which are variants of the minimax principle Let j b e the standard Hermitian n pro duct on C If s n r denote by G the Grassmannian of r dimensional linear subspaces rs n of C Then Axjx A min max j LG xL n jj xjx n The idea is to test the values of Axjx on subspaces of C Thats how Hermann Weyl proved in the following inequalities Prop osition C A B pq p q Proof The rst p oint is to understand what happ ens when you mo dify a Hermitian matrix by n another one of small rank To x notations let e e b e a basis of C made of eigenvectors n of A for the eigenvalues A A n Lemma Suppose that rank B k Then C A k Proof For reasons of dimensions the vector space generated by e e meets the kernel k of B along a nonzero unitary vector x Then C C xjx Axjx A k After adding if necessary suitable multiples of the identity we can supp ose that the eigenvalues p n of A B C are all p ositive Denote by A the Hermitian endomorphism of C dened by p A e e if i p zero if i p it dep ends on the choice of the basis of eigenvectors i i p when A A The largest eigenvalue of A A A is A A Let p p p q p q p q B B B and C A B C A B Since A B has rank at most p q the lemma implies that C C A B A B pq p q L MANIVEL Note that the main p oint in the key lemma ab ove was to nd a linear subspace in sp ecial p osition In general it is intuitively clear that the eigenvalues of C A B will dep end on the relative p osition of the eigenspaces of A and B Thats precisely the kind of information that is enco ded in Schubert varieties of Grassmannians n Schubert varieties Let V denote a complete ag in C that is a sequence of linear subspaces n V V V C i n where V has dimension i Let b e a partition inscrib ed in a r by s n r rectangle that is i a sequence of integers s We dene the Schubert cell r V fW G dim W V i for s i j s i g rs j i i and the Schubert variety X V fW G dim W V i i r g rs si i For example when has a unique non zero part k we get a special Schubert variety X V fW G W V g rs k sk We will use the notations and X when the reference ag V do es not matter The following facts are wellknown see eg The Schubert variety X is a closed subvariety of G dened lo cally by the vanishing of rs n n minors of the comp osite maps W C C V i Let W G The sequence dim W V go es from to r increasing at most by one at rs j each step So it increases strictly at exactly r values of j which we denote by j s i i with a partition inscrib ed in a r by s rectangle This proves that a G rs r s Moreover if dim W V i we clearly have s i s i hence i i si i a X In particular we have the incidence relation X X if and only if n Chose a basis v v of C adapted to the reference ag ie such that V hv v i n i i Then W admits a unique basis of the form X x v w v ij j i si i j si i j sk k i k r sjj where i r In particular is ane isomorphic to C where jj r Moreover it is easy to check that X is the Zariski closure of It follows that the Schubert cells dene a complex cellular decomp osition of the Grassmannian An immediate consequence is that the fundamental classes of the Schubert varieties the Schubert classes X where is a partition inscrib ed in a r by s rectangle form a basis of the cohomology with integers co ecients M H G Z Z rs r s Note that the Schubert classes do not dep end on the reference ag This is b ecause GLn C acts transitively on the set of complete ags And it is a general fact that if Y is a subvariety of some variety X on which a connected top ological group G acts continuously then the fundamental class of g Y do es not dep end on g G AROUND THE HORN CONJECTURE P k k q rank H G Z b e the Poincare polynomial of the Grass Exercise Let P G Z rs q rs k mannian Prove that when q is a p ower of a prime P G equals the number of p oints of the q rs grassmannian G F over the eld with q elements Deduce from this interpretation that rs q r s q q q P G q rs r s q q q q How to multiply Schubert classes Now that we know the additive structure of the coho mology ring of the Grassmannian we need to understand its multiplicative structure It was investigated in detail by mathematicians from the last century in particular from the German and Italian schools Here are the main remarkable formulas Duality Let again V b e our reference ag with an adapted basis v We dene the dual ag V by V hv v i From our explicit descriptions of a prefered basis of an element ni n i of a given Schubert cell we easily deduce the following fact if and are partitions such that jj jj r s then V and V meet transversely in a unique p oint if where s s is the complementary partition of and have empty intersection r otherwise see p or This implies that the cuppro duct r s where is the class of a p oint This means that the basis of the cohomology of the Grassman r s nian given by Schubert classes is up to complementarity selfdual relatively to Poincare duality Usually one represents a partition by its diagram as b elow this diagram has b oxes of the i ith row from top to b ottom and the complementary partition has complementary diagram after rotation in the r by s rectangle r r s s Pieris formula Let b e the class of a special Schubert variety Then k X k r s k where k denotes the space of partitions such that j j jj k and see i i i p or Giambellis formula It is a formal consequence of Pieris formula that each Schubert class can b e expressed as a determinant in sp ecial classes see p or det ij ij r i n Exercise Identifying C with its dual vector space there is a natural isomorphism G G rs sr G where Show that this isomorphism exchanges a Schubert variety X G with X sr rs is the conjugate partition of Deduce from Giambellis formula that if then k k det ij s ij i L MANIVEL r s s r There are other descriptions of the cohomology ring of the Grassmannian as a quotient of a p olynomial ring Indeed one can deduce from Giambellis formula that the sp ecial classes P k s generate H G Let z z z z One can prove s rs s k k that is the k th Chern class of the dual of the tautological vector bundle on G which has rs k rank r so that for k r p moreover this gives a complete set of relations k More precisely H G Z C h i rs s r n In principle the previous formulas are enough to compute the pro duct of any two Schu b ert classes you just need to use Giambellis formula to express in terms of sp ecial Schubert classes and then apply Pieris formula r times Of course this is not very satisfactory Little woo d and Richardson gave in a combinatorial algorithm for computing the multiplicities in the pro duct c X c can b e non zero only when c The LittlewoodRichardson rule First observe that c A skewtableau on will b e a way to ll the complement of the diagram of inside that of by p ositive integers which increase on each column from top to b ottom and do not decrease on each line from left to right It will have weight if each integer app ears exactly times in the lling Finally let w w w b e the asso ciated word obtained by reading i jj the numbers in the skewtableau line after line from right to left It is a lattice word if in every subword w w each integer j app ears at least as often as j see I i equals the number of skewtableaux on The LittlewoodRichardson coecient c of weight whose associated word is a lattice word Example In computing the pro duct the following skewtableaux do contribute and we get There exist several other combinatorial descriptions of Littlewoo dRichardson co ecients A recently discovered one is due to Knutson and Tao and has b een essential in their pro
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