Grassmann Coordinates

Size: px
Start display at page:

Download "Grassmann Coordinates"

Transcription

1 Grassmann Coordinates and tableaux Matthew Junge Autumn 2012

2 Goals 1 Describe the classical embedding G(k, n) P N. 2 Characterize the image of the embedding quadratic relations. vanishing polynomials. 3 Reinterpret in terms of varieties and ideals. 4 Application: classify representations over GL n (C).

3 What is a Grassmannian? A Grassmannian G(k, n) is the set of all k-dimensional subspaces of C n. For example, G(1, 3) = P 2 where we identify all lines. G(k, n) can be given a topology by embedding it as a subspace of P N.

4 The Embedding Fix n, k and fix a basis for C n. Let S k G(k, n) be k-dimensional subspace. Goal: Map S k to a point in P (n k) 1.

5 S k p I P (n k) 1 Let α 1,..., α k C n be a basis for S k, and let A = the corresponding k n matrix. α 1. α k Let I = i 1... i k with each 1 i j n and i 1 < i 2 < < i k. Let A I denote the k k submatrix obtained by selecting the columns with suffixes i 1,..., i k. be

6 S k p I P (n k) 1 Let α 1,..., α k C n be a basis for S k, and let A = the corresponding k n matrix. α 1. α k Let I = i 1... i k with each 1 i j n and i 1 < i 2 < < i k. Let A I denote the k k submatrix obtained by selecting the columns with suffixes i 1,..., i k. We define coordinate functions Φ I (A I ) = det A I := p I. This gives a map Φ : G(n, k) P (n k) 1 be S k (..., p I,...), I. }{{} ( k)-tuple n

7 Details About Embedding Proposition Φ is injective. Messy argument with coordinates.

8 Details About Embedding Proposition Φ is injective. Messy argument with coordinates. Proposition Φ is not surjective.

9 The Plücker Relations G(k, n) P (n k) 1. Goal: Characterize the image of G(k, n). Let X = Φ(G(k, n)). The points in X satisfy certain quadratic relations.

10 The Plücker Relations G(k, n) P (n k) 1. Goal: Characterize the image of G(k, n). Let X = Φ(G(k, n)). The points in X satisfy certain quadratic relations. Proposition The points in X do not satisfy any linear relations.

11 The Plücker Relations Theorem (Plücker Relations) Fix p X. For all 1 s n and any coordinates p I, p J with I = i 1... i k and J = j 1... j k it holds that p I p J = k p i1...i s 1 j λ i s+1...i k p j1 j λ 1 i sj λ+1...j k. λ=1

12 The Plücker Relations Theorem (Plücker Relations) Fix p X. For all 1 s n and any coordinates p I, p J with I = i 1... i k and J = j 1... j k it holds that p I p J = k p i1...i s 1 j λ i s+1...i k p j1 j λ 1 i sj λ+1...j k. λ=1 Theorem (Surjectivity Theorem) If p P N satisfies the Plücker relations then there is a k-space S k P n with coordinate p.

13 Basis Theorem I Definition Let 1 i 1,..., i k 1 n and let 1 j 1,..., j k+1 n be distinct numbers. Denote these two choices by I and J. We define a quadratic basis polynomial k+1 F IJ (P) = ( 1) λ P i1...i k 1 j λ P j1...j λ 1 j λ+1 j k+1 λ=1 with the P L inderminates.

14 Basis Theorem I Proposition For all p X and all I, J it holds that F IJ (p) = 0.

15 Basis Theorem I Proposition For all p X and all I, J it holds that F IJ (p) = 0. But what if G(p) = 0? For arbitrary homogeneuos G.

16 Basis Theorem Theorem (Basis Theorem I) If G(P) is a homogeneous polynomial in the indeterminates..., P L,... with L = l 1... l k such that G(p) = 0, p X then G(P) = I,J A IJ (P)F IJ (P), I = i 1... i k 1, J = j 1... j k+1 (1) with the F IJ quadratic basis polynomials and A IJ homogeneous polynomials in the P L.

17 Summary We can embed G(k, n) into P (n k) 1. The image consists of points satisfying certain quadratic (Plücker) relations. The set of polynomials which vanish on the image is generated by a set of quadratic polynomials.

18 Summary We can embed G(k, n) into P (n k) 1. The image consists of points satisfying certain quadratic (Plücker) relations. The set of polynomials which vanish on the image is generated by a set of quadratic polynomials. Up Next: This can all be reformulated and proven in terms of varieties and ideals in a coordinate free way.

19 Coordinate-Free Version Let E be a C-vector space, recall that ( d d ) E = E /T with T = {v 1 v d sign (σ)v σ(1) v σ(d) } d E is multilinear. d E is anticommutative.

20 Coordinate-Free Embedding ( d ) Fix S n d G(n d, n). Will map G(n d, n) P E via S n d H Sn d.

21 Coordinate-Free Embedding ( d ) Fix S n d G(n d, n). Will map G(n d, n) P E via S n d H Sn d. The kernel of the map d E d (E/S n d ) is a hyperplane H Sn d d E. Recall that P (E) is the quotient of E in which we identify all hyperplanes.

22 Polynomials on P ( d E) For any v 1,... v d E we can define a linear form on H P ( d E). d E π ( d E) /H := L For f L. ( ) d π E L (v 1 v d )(H) := (v 1 v d )(L ) (π f )(v 1 v d ) { 0 0 Products of the v 1 v d live in Sym ( ) d E.

23 Plücker Relations Theorem (The Plücker Relations/Surjectivity) The Plücker embedding is a bijection from G(n d, n) to the subvariety of P ( d E) defined by the quadratic equations (v 1 v d ) (w 1 w d ) = i 1 <i 2 < <i k (v 1 w 1 w k v d ) (v i1 v ik w k+1 w d )

24 The Basis Theorem II Theorem (The Basis Theorem II) Let Q be the ideal generated by the Plücker Relations. It holds that I(Z( Q)) = Q. Proof. We will prove that Q is prime. The Nullstellensatz immediately implies the result.

25 Proving Primality of Q. Short-Story: Goal is to show that Sym ( d E ) / Q is an integral domain. Will prove it embeds as a subring of a polynomial ring. Obtain a classification of polynomial representations over GL n (C).

26 Proving Primality of Q. Short-Story: Goal is to show that Sym ( d E ) / Q is an integral domain. Will prove it embeds as a subring of a polynomial ring. Obtain a classification of polynomial representations over GL n (C). First we need to introduce the tableaux:

27 Tableaux Let E be a C-module. For fixed n, we let λ denote a weakly decreasing partition of n, i.e. for n = 16 a partition λ could be λ = (6, 4, 4, 2) = 16. The associated tableau (also denoted λ) is

28 Constructing the Schur Module: Step 1/4 From each λ we can construct a particular C-module E λ. Start with cartesian product E λ Instead of n-tuples - put elements in boxes.

29 Constructing the Schur Module: Step 1/4 From each λ we can construct a particular C-module E λ. Start with cartesian product E λ Instead of n-tuples - put elements in boxes. If n = 5 and λ = (2, 2, 1) we have an element v E λ is written v 1 v 4 v 2 v 5 v 3

30 Constructing the Schur Module: Step 2/4 Let λ have s columns and let d i, i = 1,..., s denote the length of the i th column. For example, E λ s d i E : v v i=1 1 v 1 v 4 v 2 v 5 v 3 (v1 v 2 v 3 ) (v 4 v 5 )

31 The Quadratic Relations: Step 3/4 Let Q λ be the submodule generated by v w The sum is over all w obtained from v with an exchange between two given columns with a given subset of boxes in the right chosen column }{{} v }{{} w

32 The Schur Module: Step 4/4 s d i E λ := E /Q λ. i=1 1 1 λ = } {{ } n times then E λ = Sym n (E). 2 λ = then E λ = n E.

33 E λ in Coordinates Let e 1,..., e n be a basis for E. Fill λ with the e i. Weakly increasing across rows. Strictly increasing down columns. Each such arrangement, T, is called a standard filling. The image of this element in E λ will be denoted by e T. e 1 e 2 e 2 e 3 e 4 e 5 e 5 e 5 e T E λ

34 E λ in Coordinates Let e 1,..., e n be a basis for E. Fill λ with the e i. Weakly increasing across rows. Strictly increasing down columns. Each such arrangement, T, is called a standard filling. The image of this element in E λ will be denoted by e T. e 1 e 2 e 2 e 3 e 4 e 5 e 5 e 5 e T E λ Theorem E λ is free on the e T.

35 A New Polynomial Ring C[Z ] := C[..., Z i,j,...], i = 1,..., m j = 1,..., n For d m choose 0 i 1 i d n. Define the polynomial Z 1,i1 Z 1,id D i1...i d = det Z d,i1 Z d,id

36 A New Polynomial Ring C[Z ] := C[..., Z i,j,...], i = 1,..., m j = 1,..., n For d m choose 0 i 1 i d n. Define the polynomial Z 1,i1 Z 1,id D i1...i d = det Z d,i1 Z d,id For an arbitrary filling T of λ with the numbers {1,..., n}, Corollary D T = s i=1 D T (1,i),T (2,i),...,T (di,i), The map e T D T is an injective homomorphism E λ C[Z ] and its image D λ is free on the polynomials D T.

37 Tying it Together λ with s columns with lengths d i each occurring with multiplicity a i. E λ Sym a 1 ( d 1 E) Sym as ( d s E)/Q λ. Define S (E; d 1,..., d s ) := E λ, Q := Q λ. (a 1,...,a s) (a 1,...,a s) R := Sym a 1 ( d 1 E) Sym as ( d s E) (a 1,...,a s) R/Q = (a 1,...,a s) E λ

38 Putting it Together. We now have 1 R/Q = (a 1,...,a s) E λ. 2 E λ D λ C[Z ] under the map e T D T Proposition Q is a prime ideal. Proof. R/Q D λ C[Z ] via e T D T. D λ remains direct (requires proof) and thus is a subring. A subring of a polynomial ring is an integral domain. Q is prime.

39 What about Q? Back to G(n d, n) and Q. Corresponds to λ has columns of length d. E λ = ( d ) ( Sym a E /Q λ = Sym d ) E / Q. a a Which we just proved embeds as a subring of a polynomial ring. Hence Q is prime as a special case. Last item of business (time pending): Why is D λ direct?

40 Some Representation Theory A representation of GL(n, C) on C is a homomorphism V : GL(n, C) GL(m, C) for some m. Let X i,j : GL(n, C) C be the coordinate function with 1 i, j n. We say that a representation, V, is polynomial if there is a basis v 1,..., v m of V such that for g GL(n, C) we have gv b = a f ab (g)v a, 1 a, b n. With f ab C[X ij ] (i.e. f ab is a polynomial).

41 E λ as a Polynomial Representation Let λ = n, e T E λ acts on a matrix g GL(m, C) via the formula g e T = g i1,j 1 g im,j m e T where the sum is taken over the n m fillings of T of obtained from T by replacing the entries (j 1,..., j m ) by (i 1,..., i m ).

42 E λ as a Polynomial Representation Let λ = n, e T E λ acts on a matrix g GL(m, C) via the formula g e T = g i1,j 1 g im,j m e T where the sum is taken over the n m fillings of T of obtained from T by replacing the entries (j 1,..., j m ) by (i 1,..., i m ). Theorem As λ varies over all tableaux the E λ classify uniquely all irreducible polynomial representations of GL(n, C).

43 E λ as a Polynomial Representation Let λ = n, e T E λ acts on a matrix g GL(m, C) via the formula g e T = g i1,j 1 g im,j m e T where the sum is taken over the n m fillings of T of obtained from T by replacing the entries (j 1,..., j m ) by (i 1,..., i m ). Theorem As λ varies over all tableaux the E λ classify uniquely all irreducible polynomial representations of GL(n, C). Proposition Any sum of irreducible pairwise distinct representations is direct. D λ remains direct.

44 Conclusion 1 G(k, n) P N in coordinates and G(n d, n) P ( d E) via a coordinate free way. 2 Can classify the vanishing polynomials on the respective images. 3 All polynomial representations of GL(m, k) have the form E λ.

45 Conclusion 1 G(k, n) P N in coordinates and G(n d, n) P ( d E) via a coordinate free way. 2 Can classify the vanishing polynomials on the respective images. 3 All polynomial representations of GL(m, k) have the form E λ. A N Y Q U E S T? I O N S

Schubert Varieties. P. Littelmann. May 21, 2012

Schubert Varieties. P. Littelmann. May 21, 2012 Schubert Varieties P. Littelmann May 21, 2012 Contents Preface 1 1 SMT for Graßmann varieties 3 1.1 The Plücker embedding.................... 4 1.2 Monomials and tableaux.................... 12 1.3 Straightening

More information

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need

More information

Combinatorics for algebraic geometers

Combinatorics for algebraic geometers Combinatorics for algebraic geometers Calculations in enumerative geometry Maria Monks March 17, 214 Motivation Enumerative geometry In the late 18 s, Hermann Schubert investigated problems in what is

More information

REPRESENTATIONS OF S n AND GL(n, C)

REPRESENTATIONS OF S n AND GL(n, C) REPRESENTATIONS OF S n AND GL(n, C) SEAN MCAFEE 1 outline For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G Although

More information

Rings and groups. Ya. Sysak

Rings and groups. Ya. Sysak Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...

More information

Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

More information

COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES

COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES ROBERT M. GURALNICK AND B.A. SETHURAMAN Abstract. In this note, we show that the set of all commuting d-tuples of commuting n n matrices that

More information

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................

More information

4.2 Chain Conditions

4.2 Chain Conditions 4.2 Chain Conditions Imposing chain conditions on the or on the poset of submodules of a module, poset of ideals of a ring, makes a module or ring more tractable and facilitates the proofs of deep theorems.

More information

MODULI SPACES AND INVARIANT THEORY 5

MODULI SPACES AND INVARIANT THEORY 5 MODULI SPACES AND INVARIANT THEORY 5 1. GEOMETRY OF LINES (JAN 19,21,24,26,28) Let s start with a familiar example of a moduli space. Recall that the Grassmannian G(r, n) parametrizes r-dimensional linear

More information

A Grassmann Algebra for Matroids

A Grassmann Algebra for Matroids Joint work with Jeffrey Giansiracusa, Swansea University, U.K. Matroid (Whitney, 1935) A collection B of subsets of [n] = {1,2,...,n}, called bases, such that the basis exchange property holds: Matroid

More information

This is a closed subset of X Y, by Proposition 6.5(b), since it is equal to the inverse image of the diagonal under the regular map:

This is a closed subset of X Y, by Proposition 6.5(b), since it is equal to the inverse image of the diagonal under the regular map: Math 6130 Notes. Fall 2002. 7. Basic Maps. Recall from 3 that a regular map of affine varieties is the same as a homomorphism of coordinate rings (going the other way). Here, we look at how algebraic properties

More information

Combinatorial bases for representations. of the Lie superalgebra gl m n

Combinatorial bases for representations. of the Lie superalgebra gl m n Combinatorial bases for representations of the Lie superalgebra gl m n Alexander Molev University of Sydney Gelfand Tsetlin bases for gln Gelfand Tsetlin bases for gl n Finite-dimensional irreducible representations

More information

ADVANCED TOPICS IN ALGEBRAIC GEOMETRY

ADVANCED TOPICS IN ALGEBRAIC GEOMETRY ADVANCED TOPICS IN ALGEBRAIC GEOMETRY DAVID WHITE Outline of talk: My goal is to introduce a few more advanced topics in algebraic geometry but not to go into too much detail. This will be a survey of

More information

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then

More information

An introduction to Hodge algebras

An introduction to Hodge algebras An introduction to Hodge algebras Federico Galetto May 28, 2009 The Grassmannian Let k be a field, E a k-vector space of dimension m Define Grass(n, E) := {V E dim V = n} If {e 1,, e m } is a basis of

More information

The varieties of e-th powers

The varieties of e-th powers The varieties of e-th powers M.A. Bik Master thesis Thesis advisors: dr. R.M. van Luijk dr. R.I. van der Veen Date master exam: 25 July 2016 Mathematisch Instituut, Universiteit Leiden Acknowledgements

More information

1. Algebraic vector bundles. Affine Varieties

1. Algebraic vector bundles. Affine Varieties 0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

More information

Homework 5 M 373K Mark Lindberg and Travis Schedler

Homework 5 M 373K Mark Lindberg and Travis Schedler Homework 5 M 373K Mark Lindberg and Travis Schedler 1. Artin, Chapter 3, Exercise.1. Prove that the numbers of the form a + b, where a and b are rational numbers, form a subfield of C. Let F be the numbers

More information

Equivariant cohomology of infinite-dimensional Grassmannian and shifted Schur functions

Equivariant cohomology of infinite-dimensional Grassmannian and shifted Schur functions Equivariant cohomology of infinite-dimensional Grassmannian and shifted Schur functions Jia-Ming (Frank) Liou, Albert Schwarz February 28, 2012 1. H = L 2 (S 1 ): the space of square integrable complex-valued

More information

Representations and Linear Actions

Representations and Linear Actions Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

More information

MATH32062 Notes. 1 Affine algebraic varieties. 1.1 Definition of affine algebraic varieties

MATH32062 Notes. 1 Affine algebraic varieties. 1.1 Definition of affine algebraic varieties MATH32062 Notes 1 Affine algebraic varieties 1.1 Definition of affine algebraic varieties We want to define an algebraic variety as the solution set of a collection of polynomial equations, or equivalently,

More information

10. Noether Normalization and Hilbert s Nullstellensatz

10. Noether Normalization and Hilbert s Nullstellensatz 10. Noether Normalization and Hilbert s Nullstellensatz 91 10. Noether Normalization and Hilbert s Nullstellensatz In the last chapter we have gained much understanding for integral and finite ring extensions.

More information

ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

More information

Yangian Flags via Quasideterminants

Yangian Flags via Quasideterminants Yangian Flags via Quasideterminants Aaron Lauve LaCIM Université du Québec à Montréal Advanced Course on Quasideterminants and Universal Localization CRM, Barcelona, February 2007 lauve@lacim.uqam.ca Key

More information

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.

FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0

More information

4.4 Noetherian Rings

4.4 Noetherian Rings 4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)

More information

ABSTRACT. Department of Mathematics. interesting results. A graph on n vertices is represented by a polynomial in n

ABSTRACT. Department of Mathematics. interesting results. A graph on n vertices is represented by a polynomial in n ABSTRACT Title of Thesis: GRÖBNER BASES WITH APPLICATIONS IN GRAPH THEORY Degree candidate: Angela M. Hennessy Degree and year: Master of Arts, 2006 Thesis directed by: Professor Lawrence C. Washington

More information

Algebraic Geometry (Math 6130)

Algebraic Geometry (Math 6130) Algebraic Geometry (Math 6130) Utah/Fall 2016. 2. Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane,

More information

Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed.

Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed. Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed. Answer: Note that the first generator factors as (y

More information

Algebras. Chapter Definition

Algebras. Chapter Definition Chapter 4 Algebras 4.1 Definition It is time to introduce the notion of an algebra over a commutative ring. So let R be a commutative ring. An R-algebra is a ring A (unital as always) that is an R-module

More information

REPRESENTATION THEORY WEEK 5. B : V V k

REPRESENTATION THEORY WEEK 5. B : V V k REPRESENTATION THEORY WEEK 5 1. Invariant forms Recall that a bilinear form on a vector space V is a map satisfying B : V V k B (cv, dw) = cdb (v, w), B (v 1 + v, w) = B (v 1, w)+b (v, w), B (v, w 1 +

More information

Q N id β. 2. Let I and J be ideals in a commutative ring A. Give a simple description of

Q N id β. 2. Let I and J be ideals in a commutative ring A. Give a simple description of Additional Problems 1. Let A be a commutative ring and let 0 M α N β P 0 be a short exact sequence of A-modules. Let Q be an A-module. i) Show that the naturally induced sequence is exact, but that 0 Hom(P,

More information

Yuriy Drozd. Intriduction to Algebraic Geometry. Kaiserslautern 1998/99

Yuriy Drozd. Intriduction to Algebraic Geometry. Kaiserslautern 1998/99 Yuriy Drozd Intriduction to Algebraic Geometry Kaiserslautern 1998/99 CHAPTER 1 Affine Varieties 1.1. Ideals and varieties. Hilbert s Basis Theorem Let K be an algebraically closed field. We denote by

More information

Twisted commutative algebras and related structures

Twisted commutative algebras and related structures Twisted commutative algebras and related structures Steven Sam University of California, Berkeley April 15, 2015 1/29 Matrices Fix vector spaces V and W and let X = V W. For r 0, let X r be the set of

More information

Topics in Module Theory

Topics in Module Theory Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and constructions concerning modules over (primarily) noncommutative rings that will be needed to study

More information

Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

More information

ERRATA. Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on March 4, 2009)

ERRATA. Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on March 4, 2009) ERRATA Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on March 4, 2009) These are errata for the Third Edition of the book. Errata from previous editions have been fixed

More information

Citation Osaka Journal of Mathematics. 43(2)

Citation Osaka Journal of Mathematics. 43(2) TitleIrreducible representations of the Author(s) Kosuda, Masashi Citation Osaka Journal of Mathematics. 43(2) Issue 2006-06 Date Text Version publisher URL http://hdl.handle.net/094/0396 DOI Rights Osaka

More information

THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES

THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES Horiguchi, T. Osaka J. Math. 52 (2015), 1051 1062 THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES TATSUYA HORIGUCHI (Received January 6, 2014, revised July 14, 2014) Abstract The main

More information

Boston College, Department of Mathematics, Chestnut Hill, MA , May 25, 2004

Boston College, Department of Mathematics, Chestnut Hill, MA , May 25, 2004 NON-VANISHING OF ALTERNANTS by Avner Ash Boston College, Department of Mathematics, Chestnut Hill, MA 02467-3806, ashav@bcedu May 25, 2004 Abstract Let p be prime, K a field of characteristic 0 Let (x

More information

Math 418 Algebraic Geometry Notes

Math 418 Algebraic Geometry Notes Math 418 Algebraic Geometry Notes 1 Affine Schemes Let R be a commutative ring with 1. Definition 1.1. The prime spectrum of R, denoted Spec(R), is the set of prime ideals of the ring R. Spec(R) = {P R

More information

Representations. 1 Basic definitions

Representations. 1 Basic definitions Representations 1 Basic definitions If V is a k-vector space, we denote by Aut V the group of k-linear isomorphisms F : V V and by End V the k-vector space of k-linear maps F : V V. Thus, if V = k n, then

More information

Pure Math 764, Winter 2014

Pure Math 764, Winter 2014 Compact course notes Pure Math 764, Winter 2014 Introduction to Algebraic Geometry Lecturer: R. Moraru transcribed by: J. Lazovskis University of Waterloo April 20, 2014 Contents 1 Basic geometric objects

More information

10. Smooth Varieties. 82 Andreas Gathmann

10. Smooth Varieties. 82 Andreas Gathmann 82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

More information

Algebra Exam Syllabus

Algebra Exam Syllabus Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate

More information

Note that a unit is unique: 1 = 11 = 1. Examples: Nonnegative integers under addition; all integers under multiplication.

Note that a unit is unique: 1 = 11 = 1. Examples: Nonnegative integers under addition; all integers under multiplication. Algebra fact sheet An algebraic structure (such as group, ring, field, etc.) is a set with some operations and distinguished elements (such as 0, 1) satisfying some axioms. This is a fact sheet with definitions

More information

4. Noether normalisation

4. Noether normalisation 4. Noether normalisation We shall say that a ring R is an affine ring (or affine k-algebra) if R is isomorphic to a polynomial ring over a field k with finitely many indeterminates modulo an ideal, i.e.,

More information

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime

More information

MODEL ANSWERS TO THE FOURTH HOMEWORK

MODEL ANSWERS TO THE FOURTH HOMEWORK MODEL ANSWES TO THE OUTH HOMEWOK 1. (i) We first try to define a map left to right. Since finite direct sums are the same as direct products, we just have to define a map to each factor. By the universal

More information

LINEAR EQUATIONS WITH UNKNOWNS FROM A MULTIPLICATIVE GROUP IN A FUNCTION FIELD. To Professor Wolfgang Schmidt on his 75th birthday

LINEAR EQUATIONS WITH UNKNOWNS FROM A MULTIPLICATIVE GROUP IN A FUNCTION FIELD. To Professor Wolfgang Schmidt on his 75th birthday LINEAR EQUATIONS WITH UNKNOWNS FROM A MULTIPLICATIVE GROUP IN A FUNCTION FIELD JAN-HENDRIK EVERTSE AND UMBERTO ZANNIER To Professor Wolfgang Schmidt on his 75th birthday 1. Introduction Let K be a field

More information

Math 215B: Solutions 1

Math 215B: Solutions 1 Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an

More information

EXTERIOR POWERS KEITH CONRAD

EXTERIOR POWERS KEITH CONRAD EXTERIOR POWERS KEITH CONRAD 1. Introduction Let R be a commutative ring. Unless indicated otherwise, all modules are R-modules and all tensor products are taken over R, so we abbreviate R to. A bilinear

More information

EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TWO: DEFINITIONS AND BASIC PROPERTIES

EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TWO: DEFINITIONS AND BASIC PROPERTIES EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TWO: DEFINITIONS AND BASIC PROPERTIES WILLIAM FULTON NOTES BY DAVE ANDERSON 1 For a Lie group G, we are looking for a right principal G-bundle EG BG,

More information

Math 203A, Solution Set 6.

Math 203A, Solution Set 6. Math 203A, Solution Set 6. Problem 1. (Finite maps.) Let f 0,..., f m be homogeneous polynomials of degree d > 0 without common zeros on X P n. Show that gives a finite morphism onto its image. f : X P

More information

5 Quiver Representations

5 Quiver Representations 5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )

More information

BEZOUT S THEOREM CHRISTIAN KLEVDAL

BEZOUT S THEOREM CHRISTIAN KLEVDAL BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this

More information

ne varieties (continued)

ne varieties (continued) Chapter 2 A ne varieties (continued) 2.1 Products For some problems its not very natural to restrict to irreducible varieties. So we broaden the previous story. Given an a ne algebraic set X A n k, we

More information

h r t r 1 (1 x i=1 (1 + x i t). e r t r = i=1 ( 1) i e i h r i = 0 r 1.

h r t r 1 (1 x i=1 (1 + x i t). e r t r = i=1 ( 1) i e i h r i = 0 r 1. . Four definitions of Schur functions.. Lecture : Jacobi s definition (ca 850). Fix λ = (λ λ n and X = {x,...,x n }. () a α = det ( ) x α i j for α = (α,...,α n ) N n (2) Jacobi s def: s λ = a λ+δ /a δ

More information

12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n 12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

More information

Finite Fields. [Parts from Chapter 16. Also applications of FTGT]

Finite Fields. [Parts from Chapter 16. Also applications of FTGT] Finite Fields [Parts from Chapter 16. Also applications of FTGT] Lemma [Ch 16, 4.6] Assume F is a finite field. Then the multiplicative group F := F \ {0} is cyclic. Proof Recall from basic group theory

More information

Algebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra

Algebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial

More information

Rings and Fields Theorems

Rings and Fields Theorems Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)

More information

A BASIS OF BIDETERMINANTS FOR THE COORDINATE RING OF THE ORTHOGONAL GROUP

A BASIS OF BIDETERMINANTS FOR THE COORDINATE RING OF THE ORTHOGONAL GROUP A BASIS OF BIDETERMINANTS FOR THE COORDINATE RING OF THE ORTHOGONAL GROUP GERALD CLIFF Abstract. We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n, K), where

More information

Algebraic structures I

Algebraic structures I MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one

More information

Topics in linear algebra

Topics in linear algebra Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over

More information

Algebraic varieties. Chapter A ne varieties

Algebraic varieties. Chapter A ne varieties Chapter 4 Algebraic varieties 4.1 A ne varieties Let k be a field. A ne n-space A n = A n k = kn. It s coordinate ring is simply the ring R = k[x 1,...,x n ]. Any polynomial can be evaluated at a point

More information

Algebraic Geometry: MIDTERM SOLUTIONS

Algebraic Geometry: MIDTERM SOLUTIONS Algebraic Geometry: MIDTERM SOLUTIONS C.P. Anil Kumar Abstract. Algebraic Geometry: MIDTERM 6 th March 2013. We give terse solutions to this Midterm Exam. 1. Problem 1: Problem 1 (Geometry 1). When is

More information

POLYNOMIAL BEHAVIOUR OF KOSTKA NUMBERS

POLYNOMIAL BEHAVIOUR OF KOSTKA NUMBERS POLYNOMIAL BEHAVIOUR OF KOSTKA NUMBERS DINUSHI MUNASINGHE Abstract. Given two standard partitions λ + = (λ + 1 λ+ s ) and λ = (λ 1 λ r ) we write λ = (λ +, λ ) and set s λ (x 1,..., x t ) := s λt (x 1,...,

More information

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

More information

Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman

Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 17, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Integral Domains and Fraction Fields 0.1.1 Theorems Now what we are going

More information

Bounded-rank tensors are defined in bounded degree. J. Draisma and J. Kuttler

Bounded-rank tensors are defined in bounded degree. J. Draisma and J. Kuttler Bounded-rank tensors are defined in bounded degree J. Draisma and J. Kuttler REPORT No. 15, 2010/2011, spring ISSN 1103-467X ISRN IML-R- -15-10/11- -SE+spring BOUNDED-RANK TENSORS ARE DEFINED IN BOUNDED

More information

4.1. Paths. For definitions see section 2.1 (In particular: path; head, tail, length of a path; concatenation;

4.1. Paths. For definitions see section 2.1 (In particular: path; head, tail, length of a path; concatenation; 4 The path algebra of a quiver 41 Paths For definitions see section 21 (In particular: path; head, tail, length of a path; concatenation; oriented cycle) Lemma Let Q be a quiver If there is a path of length

More information

Cell decompositions and flag manifolds

Cell decompositions and flag manifolds Cell decompositions and flag manifolds Jens Hemelaer October 30, 2014 The main goal of these notes is to give a description of the cohomology of a Grassmannian (as an abelian group), and generalise this

More information

Algebraic Varieties. Chapter Algebraic Varieties

Algebraic Varieties. Chapter Algebraic Varieties Chapter 12 Algebraic Varieties 12.1 Algebraic Varieties Let K be a field, n 1 a natural number, and let f 1,..., f m K[X 1,..., X n ] be polynomials with coefficients in K. Then V = {(a 1,..., a n ) :

More information

Math 797W Homework 4

Math 797W Homework 4 Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

More information

Littlewood Richardson polynomials

Littlewood Richardson polynomials Littlewood Richardson polynomials Alexander Molev University of Sydney A diagram (or partition) is a sequence λ = (λ 1,..., λ n ) of integers λ i such that λ 1 λ n 0, depicted as an array of unit boxes.

More information

ADVANCED COMMUTATIVE ALGEBRA: PROBLEM SETS

ADVANCED COMMUTATIVE ALGEBRA: PROBLEM SETS ADVANCED COMMUTATIVE ALGEBRA: PROBLEM SETS UZI VISHNE The 11 problem sets below were composed by Michael Schein, according to his course. Take into account that we are covering slightly different material.

More information

Geometry 2: Manifolds and sheaves

Geometry 2: Manifolds and sheaves Rules:Exam problems would be similar to ones marked with! sign. It is recommended to solve all unmarked and!-problems or to find the solution online. It s better to do it in order starting from the beginning,

More information

Again we return to a row of 1 s! Furthermore, all the entries are positive integers.

Again we return to a row of 1 s! Furthermore, all the entries are positive integers. Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Updated: April 4, 207 Examples. Conway-Coxeter frieze pattern Frieze is the wide central section part of an entablature, often seen in Greek temples,

More information

V (f) :={[x] 2 P n ( ) f(x) =0}. If (x) ( x) thenf( x) =

V (f) :={[x] 2 P n ( ) f(x) =0}. If (x) ( x) thenf( x) = 20 KIYOSHI IGUSA BRANDEIS UNIVERSITY 2. Projective varieties For any field F, the standard definition of projective space P n (F ) is that it is the set of one dimensional F -vector subspaces of F n+.

More information

Representation Theory

Representation Theory Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section II 19I 93 (a) Define the derived subgroup, G, of a finite group G. Show that if χ is a linear character

More information

Notas de Aula Grupos Profinitos. Martino Garonzi. Universidade de Brasília. Primeiro semestre 2018

Notas de Aula Grupos Profinitos. Martino Garonzi. Universidade de Brasília. Primeiro semestre 2018 Notas de Aula Grupos Profinitos Martino Garonzi Universidade de Brasília Primeiro semestre 2018 1 Le risposte uccidono le domande. 2 Contents 1 Topology 4 2 Profinite spaces 6 3 Topological groups 10 4

More information

Math 594, HW2 - Solutions

Math 594, HW2 - Solutions Math 594, HW2 - Solutions Gilad Pagi, Feng Zhu February 8, 2015 1 a). It suffices to check that NA is closed under the group operation, and contains identities and inverses: NA is closed under the group

More information

Picard-Vessiot theory, algebraic groups and group schemes

Picard-Vessiot theory, algebraic groups and group schemes Picard-Vessiot theory, algebraic groups and group schemes Jerald J. Kovacic Department of Mathematics The City College of The City University of New York New York, NY 003 email: jkovacic@verizon.net URL:

More information

ALGEBRAIC GEOMETRY (NMAG401) Contents. 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30

ALGEBRAIC GEOMETRY (NMAG401) Contents. 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30 ALGEBRAIC GEOMETRY (NMAG401) JAN ŠŤOVÍČEK Contents 1. Affine varieties 1 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30 1. Affine varieties The basic objects

More information

The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6

The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6 The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6 A Dissertation in Mathematics by Evgeny Mayanskiy c 2013 Evgeny Mayanskiy Submitted in Partial

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F -vector space or simply a vector space

More information

ERRATA. Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on February 14, 2018)

ERRATA. Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on February 14, 2018) ERRATA Abstract Algebra, Third Edition by D. Dummit and R. Foote (most recently revised on February 14, 2018) These are errata for the Third Edition of the book. Errata from previous editions have been

More information

Math 530 Lecture Notes. Xi Chen

Math 530 Lecture Notes. Xi Chen Math 530 Lecture Notes Xi Chen 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca 1991 Mathematics Subject Classification. Primary

More information

Problem 1.1. Classify all groups of order 385 up to isomorphism.

Problem 1.1. Classify all groups of order 385 up to isomorphism. Math 504: Modern Algebra, Fall Quarter 2017 Jarod Alper Midterm Solutions Problem 1.1. Classify all groups of order 385 up to isomorphism. Solution: Let G be a group of order 385. Factor 385 as 385 = 5

More information

The Proj Construction

The Proj Construction The Proj Construction Daniel Murfet May 16, 2006 Contents 1 Basic Properties 1 2 Functorial Properties 2 3 Products 6 4 Linear Morphisms 9 5 Projective Morphisms 9 6 Dimensions of Schemes 11 7 Points of

More information

RINGS: SUMMARY OF MATERIAL

RINGS: SUMMARY OF MATERIAL RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered

More information

k k would be reducible. But the zero locus of f in A n+1

k k would be reducible. But the zero locus of f in A n+1 Math 145. Bezout s Theorem Let be an algebraically closed field. The purpose of this handout is to prove Bezout s Theorem and some related facts of general interest in projective geometry that arise along

More information

The Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases

The Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases arxiv:8.976v [math.rt] 3 Dec 8 The Gelfand-Tsetlin Realisation of Simple Modules and Monomial Bases Central European University Amadou Keita (keita amadou@student.ceu.edu December 8 Abstract The most famous

More information

ABSTRACT NONSINGULAR CURVES

ABSTRACT NONSINGULAR CURVES ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a

More information

The Littlewood-Richardson Rule

The Littlewood-Richardson Rule REPRESENTATIONS OF THE SYMMETRIC GROUP The Littlewood-Richardson Rule Aman Barot B.Sc.(Hons.) Mathematics and Computer Science, III Year April 20, 2014 Abstract We motivate and prove the Littlewood-Richardson

More information

Modular branching rules and the Mullineux map for Hecke algebras of type A

Modular branching rules and the Mullineux map for Hecke algebras of type A Modular branching rules and the Mullineux map for Hecke algebras of type A 1 Introduction Jonathan Brundan Department of Mathematics University of Oregon Eugene, OR 97403-1222 USA E-mail: brundan@darkwing.uoregon.edu

More information

Combinatorial Structures

Combinatorial Structures Combinatorial Structures Contents 1 Permutations 1 Partitions.1 Ferrers diagrams....................................... Skew diagrams........................................ Dominance order......................................

More information

Kostka multiplicity one for multipartitions

Kostka multiplicity one for multipartitions Kostka multiplicity one for multipartitions James Janopaul-Naylor and C. Ryan Vinroot Abstract If [λ(j)] is a multipartition of the positive integer n (a sequence of partitions with total size n), and

More information