Outline I. Flags what is a flag? who cares? II. (commutative) - - PowerPoint PPT Presentation

PO-set paths and q-commuting minors

Aaron Lauve LaCIM - UQAM

S´ eminaire de combinatoire et d’informatique math´ ematique le 7 avril 2006

http://www.lacim.uqam.ca/∼lauve (“Research”) lauve@lacim.uqam.ca

Outline

• I. Flags
• what is a flag?
• who cares?
• II. (commutative) Generic matrices
• column-minor identities
• homogeneous coordinate ring of the flag variety
• III. q-Generic matrices
• row-quantum-minor identities
• quantum flag variety
• IV. (noncommutative) Generic matrices
• row-quasi-minor identities
• specializations
• V. q-Commuting minors
• “missing” relations
• PO-set paths

1

I. Flags

• Fix: an integer n > 1 and a vector space V = Cn.
• Fix: a sequence of integers λ : n ≥ λ1 > λ2 > · · · > λs ≥ 0.

Definition (Flag). A flag Φ of shape λ is a chain of subspaces

Φ : 0 ⊆ V1 V2 · · · Vs ⊆ V

satisfying λi = codim Vi = n − dim Vi. Denote the collection of all flags of shape λ by

Fl(λ).

2

I. Flags

• Fix: an integer n > 1 and a vector space V = Cn.
• Fix: a sequence of integers λ : n ≥ λ1 > λ2 > · · · > λs ≥ 0.

Definition (Flag). A flag Φ of shape λ is a chain of subspaces

Φ : 0 ⊆ V1 V2 · · · Vs ⊆ V

satisfying λi = codim Vi = n − dim Vi. Denote the collection of all flags of shape λ by

Fl(λ).

• Example. Taking n = 6 and λ = (5, 3, 2),

Φ : span {v1} ⊆ span {v1, v2, v3} ⊆ span {v1, v2, v3, v4}

is a flag when v1, . . . , v4 are linearly independent.

3

I. Flags

• Fix: an integer n > 1 and a vector space V = Cn.
• Fix: a sequence of integers λ : n ≥ λ1 > λ2 > · · · > λs ≥ 0.

Definition (Flag). A flag Φ of shape λ is a chain of subspaces

Φ : 0 ⊆ V1 V2 · · · Vs ⊆ V

satisfying λi = codim Vi = n − dim Vi. Denote the collection of all flags of shape λ by

Fl(λ).

• Example. Taking n = 6 and λ = (5, 3, 2),

Φ : span {v1} ⊆ span {v1, v2, v3} ⊆ span {v1, v2, v3, v4}

is a flag when v1, . . . , v4 are linearly independent.

• Choose a basis B for V and express Φ as a matrix A(Φ). . .

4

I. Flags

• Φ : span {v1} ⊆ span {v1, v2, v3} ⊆ span {v1, v2, v3, v4}

A(Φ) =        a11 a12 a13 a14 a15 a16 a21 a22 a23 a24 a25 a26 a31 a32 a33 a34 a35 a36 a41 a42 a43 a44 a45 a46        V1 V2 V2 V3

5

I. Flags

• Φ : span {v1} ⊆ span {v1, v2, v3} ⊆ span {v1, v2, v3, v4}

A(Φ) =        a11 a12 a13 a14 a15 a16 a21 a22 a23 a24 a25 a26 a31 a32 a33 a34 a35 a36 a41 a42 a43 a44 a45 a46        V1 V2 V2 V3

6

I. Flags

• Φ : span {v1} ⊆ span {v1, v2, v3} ⊆ span {v1, v2, v3, v4}

A(Φ) =        a11 a12 a13 a14 a15 a16 a21 a22 a23 a24 a25 a26 a31 a32 a33 a34 a35 a36 a41 a42 a43 a44 a45 a46        V1 V2 V2 V3

7

I. Flags

• Φ : span {v1} ⊆ span {v1, v2, v3} ⊆ span {v1, v2, v3, v4}

A(Φ) =        a11 a12 a13 a14 a15 a16 a21 a22 a23 a24 a25 a26 a31 a32 a33 a34 a35 a36 a41 a42 a43 a44 a45 a46        V1 V2 V2 V3

• Unique up to change of basis! . . . multiplying by Pλ =

       ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗       

• n the left.

8

I. Who Cares?

A(Φ) =         a11 a12 · · · · · · a1n a21 a22 a2n

. . . . . . . . .

ad1 ad2 · · · · · · adn         (d = n − λs)

Representation Theory Notice that GLn(C) (and its many important subgroups) permutes

Fl(λ) by right-multiplication.

9

I. Who Cares?

A(Φ) =   a11 a12 a13 a14 a21 a22 a23 a24  

Topology/Algebraic Geometry The space Fl(λ) is a (projective) algebraic variety and a CW-complex, described via Gaussian elimination (G.E.). Consider the case n = 4, λ = (2):

• G.E.

− →

Open cells:

µ = (00)

• (10)
• (20)
• (11)
• (21)
• (22)
• 10

I. Who Cares?

A(Φ) =   a11 a12 a13 a14 a21 a22 a23 a24  

Topology/Algebraic Geometry The space Fl(λ) is a (projective) algebraic variety and a CW-complex, described via Gaussian elimination. Consider the case n = 4, λ = (2): Schubert cells Ωµ:

µ = (00)

• (10)
• (20)
• (11)
• (21)
• (22)
• Partitions µ with |λ| = 2 parts and part-size at most n − 2 = 2.

Combinatorics The classes [Ωµ] in the cohomology ring H•(Fl(λ)) are Schur polynomials!!

11

II. (commutative) Generic Matrices

• Definition. Let X = (xij) be an n × n matrix of commuting indeterminants. Call X a

generic matrix (coordinate functions for a “generic point” in Cn2).

• Definition. For I, J ∈ [n]d, put XI,J =

        xi1j1 xi1j2 · · · xi1jd xi2j1 xi2j2 · · · xi2jd

. . . . . . . . .

xidj1 xidj2 · · · xidjd        

. Let XJ denote the special case I = (1, 2, . . . , d) (take the first d rows of X). Consider the column-minors of shape λ:

M(λ) = {[ [I] ] := det XI : n − |I| ∈ λ} .

Problem: Describe the relations R among the minors M(λ).

12

II. (commutative) Generic Matrices

Answer (Schur ‘01, Hodge ‘43): The minors M(λ) satisfy the Young symmetry relations

0 =

• Λ⊆L

|Λ|=r

(−1)ℓ(L\Λ|Λ)[ [L \ Λ] ][ [Λ|M] ]

(YL,M) for any 1 ≤ r and any L, M ⊆ [n] with |M| + r ≤ |L| − r ∈ n − λ. Moreover, writing M(λ) = {I1, . . . IN}, if F(Z1, . . . , ZN) is a polynomial which is zero on substitution Zi → [

[Ii] ], then F is algebraically dependent on the (YL,M).

Example:

[ [12] ][ [34] ] − [ [13] ][ [24] ] + [ [23] ][ [14] ] = 0.

Non-example: (Sylvester’s Identity)

(det A123,123) (det A2,2) = (det A12,12) (det A23,23) − (det A12,23) (det A23,12) .

13

III.

q-Generic Matrices

• Definition. Call a matrix X q-generic if every 2 × 2 submatrix

2 6 6 6 6 4

a b c d

3 7 7 7 7 5 satisfies:

(←) ba = q ab dc = q cd ( ↑ ) ca = q ac db = q bd (ր) cb = bc (տ) da = ad + (q − q−1) bc

Define the quantum determinant by

detqXI,J =

• σ∈Sd

(−q)−ℓ(σ)xi1jσ(1) xi2jσ(2) · · · xidjσ(d)

for I, J ∈ [n]d, and let [

[J] ] now denote detqXJ.

Problem: Describe the relations R among the minors Mq(λ).

14

III.

q-Generic Matrices

Answer (Taft-Towber ‘91): The set R is (algebraically) generated by the relations below:

q-Alternating: (∀I ∈ [n]d) [ [I] ] = (−q)−ℓ(σ)[ [σI] ] if σ “straightens” I. q-Young symmetry: (∀L, M ⊆ [n], r > 0) s.t. |M| + r ≤ |L| − r 0 =

• Λ⊂L,|Λ|=r

(−q)−ℓ(L\Λ|Λ)[ [L \ Λ] ][ [Λ | M] ] .

(YL,M)

q-Straightening: (∀I, J [n]) s.t. |J| < |I| [ [J] ][ [I] ] =

• Λ⊆I,|Λ|=|J|

(−q)+ℓ(Λ|I\Λ)[ [J|I \ Λ] ][ [Λ] ] .

(SJ,I) Example:

[ [12] ][ [34] ] − 1

q[

[13] ][ [24] ] + 1

q2 [

[23] ][ [14] ] = 0.

Non-examples: (cf. Goodearl, ‘05)

15

IV. (noncommutative) Generic Matrices

• Now let X be a matrix of noncommuting variables.

Definition (Gelfand-Retakh ‘91). The (ij)-quasideterminant |X|ij is defined whenever Xij is invertible, and in that case,

|X|ij =

• 16

IV. (noncommutative) Generic Matrices

• Now let X be a matrix of noncommuting variables.

Definition (Gelfand-Retakh ‘91). The (ij)-quasideterminant |X|ij is defined whenever Xij is invertible, and in that case,

|X|ij =

• =

− ·

−1 ·

• 2 × 2 Example:

|A|12 =

• a11 a12

a21 a22

• = a12 − a11a−1

21 a22.

17

IV. (noncommutative) Generic Matrices

• It is better to take ratios of quasideterminants as “column-minor” replacements
• Definition. Given an n × n matrix X and a partition λ, the column-minors are given by

Mquasi(λ) =

• pK

ij (X) := |Xi∪K|−1 di |Xj∪K|dj

• i, j ∈ [n], K ⊆ [n] \ i, n − 1 − |K| ∈ λ
• Quasi-Pl¨

ucker Relations (Gelfand-Retakh ‘97, L. ’04): If L, M ⊆ [n], i ∈ [n] \ M,

|M| ≤ |L| − 1, then: 1 =

• j∈L

pM

ij (X)pL\j ji (X) .

(Pi,L,M) Problem: Describe the relations R satisfied by Mquasi

• Need to expand search to rational expressions, not just algebraic ones.
• some are known. . . maybe all?

18

IV. (noncommutative) Generic Matrices

Why study these matrices?

• May study “all” the deformations of the commutative generic matrix at once!
• X → commutative generic:

pK

ij → ratio of two determinants [

[i ∪ K] ]−1[ [j ∪ K] ]

• X → q-generic:

pK

ij → ratio of two quantum determinants [

[i ∪ K] ]−1[ [j ∪ K] ]

Question: How much of R(M) or R(Mq) may be recovered on specializing X and

R(Mquasi)?

Answer: For commutative case, all. For q-generic case, a lot. . .

19

V. “Missing” Relations

• While showing R(Mquasi) ⇒ (YL,M) for quantum setting, we discover something “new:”
• [

[I] ] and [ [J] ] often q-commute [Krob-Leclerc ‘95].

• Definition. Given two subsets I, J ⊆ [n], we say J surrounds I, written J I, if (i)

|J| ≤ |I|, and (ii) there exist disjoint subsets ∅ ⊆ J ′, J′′ ⊆ J such that:

• a. J \ I = J′ ˙

∪ J′′,

• b. j′ < i for all j′ ∈ J′ and i ∈ I \ J,
• c. i < j′′ for all i ∈ I \ J and j′′ ∈ J′,

Theorem (q-Commuting Minors). If the subsets I, J ⊆ [n] satisfy J I, the quantum minors [

[J] ] and [ [I] ] q-commute. Specifically, [ [J] ][ [I] ] = q|J′′|−|J′|[ [I] ][ [J] ] .

(CJ,I)

20

V. Finding the Missing Relations

Theorem (L, ‘06). Given I, J ⊆ [n]. If J I, then the sum CJ,I may be written as a

Z[q, q−1]-linear combination of the sums SJ,I and YI∪(J\K),K ( ∅ ⊆ K J ). The

coefficients involved are described in terms of path-weighs in a directed graph. Open Question (converse): Give an easy proof that only if J I can CJ,I be so written. [Leclerc-Zelevinsky ‘98]

21

V. Example 1

Writing C1,234 in terms of S1,234 and YL,M.

C1,234 f1f234 −q−1f234f1 S1,234 f1f234 −q2f123f4 +q1f124f3 −q0f134f2 q2 Y1234,∅ f123f4 −q−1f124f3 +q−2f134f2 −q−3f234f1

22

V. Example 1

Writing C1,234 in terms of S1,234 and YL,M.

C1,234 f1f234 −q−1f234f1 S1,234 f1f234 −q2f123f4 +q1f124f3 −q0f134f2 q2 Y1234,∅ q2f123f4 −q1 f124f3 +q0 f134f2 −q−1f234f1 C1,234 = S1,234 + q2Y1234,∅

23

V. Example 2

Writing C14,23 in terms of YL,M.

[14][23] [24][13] [34][12] [12][34] [13][24] [23][14] C14,23 1 −1 Y1234,∅ q−2 −q−3 q−4 1 −q−1 q−2 Y234,1 q−2 −q−3 −q−1 Y123,4 1 −q−1 q−2

24

V. Example 2

Writing C14,23 in terms of YL,M.

[14][23] [24][13] [34][12] [12][34] [13][24] [23][14] C14,23 1 −1 Y1234,∅ q−2 −q−3 q−4 1 −q−1 q−2 Y234,1 q−2 −q−3 −q−1 Y123,4 1 −q−1 q−2

25

V. Example 2

Writing C14,23 in terms of YL,M.

[14][23] [24][13] [34][12] [12][34] [13][24] [23][14] C14,23 1 −1 Y1234,∅ q−2 −q−3 q−4 1 −q−1 q−2 Y234,1 q−2 −q−3 −q−1 Y123,4 1 −q−1 q−2

26

V. Example 2

Writing C14,23 in terms of YL,M.

[14][23] [24][13] [34][12] [12][34] [13][24] [23][14] C14,23 1 −1 Y1234,∅ q−2 −q−3 q−4 1 −q−1 q−2 Y234,1 q−2 −q−3 −q−1 Y123,4 1 −q−1 q−2

27

V. Example 2

Writing C14,23 in terms of YL,M.

[14][23] [24][13] [34][12] [12][34] [13][24] [23][14] C14,23 1 −1 Y1234,∅ q−2 −q−3 q−4 1 −q−1 q−2 Y234,1 q−2 −q−3 −q−1 Y123,4 1 −q−1 q−2

28

V. Example 3

Writing C156,234 in terms of YL,M.

e∅ e1 e5 e6 e15 e16 e56 e156 y∅ 1 α1

∅

α5

∅

α6

∅

α15

∅

α16

∅

α56

∅

α156

∅

y1 1 α15

1

α16

1

α156

1

y5 1 α15

5

α56

5

α156

5

y6 1 α16

6

α56

6

α156

6

y15 1 α156

15

y16 1 α156

16

y56 1 α156

56

29

V. Example 3

Writing C156,234 in terms of YL,M.

e∅ e1 e5 e6 e15 e16 e56 e156 y∅ 1 α1

∅

α5

∅

α6

∅

α15

∅

α16

∅

α56

∅

α156

∅

y1 1 α15

1

α16

1

α156

1

y5 1 α15

5

α56

5

α156

5

y6 1 α16

6

α56

6

α156

6

y15 1 α156

15

y16 1 α156

16

y56 1 α156

56

• Perform Gaussian Elimination!!
• Hope that what remains on the first row is 1 and (−q)θq2−1

30

V. Example 3

Writing C156,234 in terms of YL,M.

e∅ e1 e5 e6 e15 e16 e56 e156 y∅ 1 α1

∅

α5

∅

α6

∅

α15

∅

α16

∅

α56

∅

α156

∅

y1 1 α15

1

α16

1

α156

1

y5 1 α15

5

α56

5

α156

5

y6 1 α16

6

α56

6

α156

6

y15 1 α156

15

y16 1 α156

16

y56 1 α156

56

• Perform Gaussian Elimination!!
• In general, we get:

coeff

• eJ

= αJ

∅ −

• ∅AJ

αA

∅ αJ A +

• ∅ABJ

αA

∅ αB AαJ B − · · ·

31

V. Example 3

A directed graph with edge-weights Γ(J; I) associated to J = 156 and I = 234.

56 156 6 16 5 15

α156

15

∅

α1

∅

1

α15

1

− →

α1

∅α15 1 α156 15 = −q−5

32

V. Example 3

A directed graph with edge-weights Γ(J; I) associated to J = 156 and I = 234.

56 156 6 16 5 15

α156

15

∅

α1

∅

1

α15

1

− →

α1

∅α15 1 α156 15 = −q−5

33

V. Example 3

A directed graph with edge-weights Γ(J; I) associated to J = 156 and I = 234.

56 156 6 16 5 15

α156

15

∅

α1

∅

1

α15

1

− →

α1

∅α15 1 α156 15 = −q−5

αB

A

:= (−q)−ℓ(J\B|B\A)−ℓ(B\A|A)+{2|J\B|−|I|}·|(B\A)∩J ′| αB

AαC B

=

• (−q)2ℓ((B\A)∩J′|C\B)−2ℓ(C\B|(B\A)∩J′′)

αC

A

34

V. PO-set paths

Let J = J′ ∪ J′′ = {j′

1, . . . , j′ r′} ∪ {j′′ 1, . . . , j′′ r′′}. We consider certain paths in Γ(J; I):

P0 =

• (A1, A2, . . . , Ap) | Ai ⊆ J s.t. ∅ A1 A2 · · · Ap J
• and P = P0 ∪ ˆ

0 ∪ ˆ 1, where ˆ 0 = (∅), and ˆ 1 = ({j′′

1}, {j′′ 1, j′′ 2}, . . . , J′′, {j′ r′, . . . , j′′ r′′}, . . . , {j′ 2, . . . , j′′ r′′}, J).

The weight α(π) of a path π = (A1, . . . , Ap) ∈ P is the product of edge weights of the augmented path (∅, π, J):

αA1

∅

· αA2

A1 · · · αAp Ap−1 · αJ Ap.

• Definition. For K ⊆ J, define mM(K) :=

   min(K ∩ J′)

if K ∩ J′ = ∅

max(K ∩ J′′)

• therwise.

For any path π = (A1, . . . , Ap), put A0 = ∅ and Ap+1 = J.

35

V. PO-set paths

• Definition. Call ˆ

1 regular and ˆ 0 irregular. Call a path (A1, . . . , Ap) ∈ P0 regular (or regular

at position i0), if (∃i0) (1 ≤ i0 ≤ p) satisfying: (a) |Ai| = i (∀ 1 ≤ i ≤ i0); (b) Ai0 \ Ai0−1 = mM(Ai0+1 \ Ai0−1). A path in P0 is irregular if it is nowhere regular.

• Proposition. The regular and irregular subsets of P are equinumerous.

A bijection ℘. Given an irregular path π = (A1, . . . , Ap) ∈ P0, we insert a new set B so that ℘(π) is regular at B:

• 1. Find the unique i0 satisfying: (|Ai| = i

∀i ≤ i0) ∧ (|Ai0+1| > i0 + 1).

• 2. Compute b = mM(Ai0+1 \ Ai0)
• 3. Put B = Ai0 ∪ {b}.
• 4. Define ℘(π) := (A1, . . . , Ai0, B, Ai0+1, . . . , Ap).

Finally, put ℘(ˆ

0) = ({j1}).

• Proposition. The bijection ℘ is path-weight preserving and α(℘−1(ˆ

1)) = (−q)θq|J′′|−|J′| .

36

56 156 6 16 5 15 ∅ 1

56 156 6 16 5 15 ∅ 1