KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The Atlas of Lie Groups andRepresentations project and software

Marc van Leeuwen

Laboratoire de Mathématiques et ApplicationsUniversité de Poitiers

Friday 23 Arpril 2010 / NMC 2010, Nijmegen

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Overview

1 The Atlas of Lie Groups and Representations project

2 The atlas software

3 A huge computation of KLV polynomials

4 Further developments

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Overview

1 The Atlas of Lie Groups and Representations project

2 The atlas software

3 A huge computation of KLV polynomials

4 Further developments

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The Atlas project

Field: Real reductive Lie groups (maybe non compact),and their (∞-dimensional) representationsGoal: get explicit data, implement modern methodsExamples: real forms, KLV polynomials, restrictions ofrepresentations to (complexified compact subgroup) KUltimate goal: unitary dual of any real reductive group

Some dates:

2002, Initial contacts2004. First NSF grantNov. 2005, atlas implements up to “KLV polynomials”10 November 2006, Death of Fokko du Cloux8 January 2007, KLV polynomials for E8(R) computed2010. Second NSF grant

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The Atlas project

Field: Real reductive Lie groups (maybe non compact),and their (∞-dimensional) representationsGoal: get explicit data, implement modern methodsExamples: real forms, KLV polynomials, restrictions ofrepresentations to (complexified compact subgroup) KUltimate goal: unitary dual of any real reductive group

Some dates:

2002, Initial contacts2004. First NSF grantNov. 2005, atlas implements up to “KLV polynomials”10 November 2006, Death of Fokko du Cloux8 January 2007, KLV polynomials for E8(R) computed2010. Second NSF grant

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Members of the Atlas project

Jeffrey Adams Alessandra PantanoDan Barbasch Annegret PaulBirne Binegar Patrick PoloBill Casselman Siddhartha SahiDan Ciubotaru Susana SalamancaFokko du Cloux John StembridgeScott Crofts Peter TrapaTatiana Howard David VoganMarc van Leeuwen Wai-Ling YeeAlfred Noel Jiu-Kang Yu

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Members of the Atlas project

Jeffrey Adams Alessandra PantanoDan Barbasch Annegret PaulBirne Binegar Patrick PoloBill Casselman Siddhartha SahiDan Ciubotaru Susana SalamancaFokko du Cloux John StembridgeScott Crofts Peter TrapaTatiana Howard David VoganMarc van Leeuwen Wai-Ling YeeAlfred Noel Jiu-Kang Yu

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The atlas software within the project

Development of software is central to the project

Main role of computation: implementing existing theory

Theory gives description in combinatorial terms, but rathercomplicated. Except for very small cases, details ofcombinatorial structures were not known

Example: “one-sided parameter set” K\G/BSet of double cosets corresponds to graph-likecombinatorial structure; generated algorithmically

For split group of type E8, even size of K\G/B was notapproximately known.Size is 320 206; only upper bound 696 729 600 was known

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The atlas software within the project

Development of software is central to the project

Main role of computation: implementing existing theory

Theory gives description in combinatorial terms, but rathercomplicated. Except for very small cases, details ofcombinatorial structures were not known

Example: “one-sided parameter set” K\G/BSet of double cosets corresponds to graph-likecombinatorial structure; generated algorithmically

For split group of type E8, even size of K\G/B was notapproximately known.Size is 320 206; only upper bound 696 729 600 was known

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The atlas software within the project

Development of software is central to the project

Main role of computation: implementing existing theory

Theory gives description in combinatorial terms, but rathercomplicated. Except for very small cases, details ofcombinatorial structures were not known

Example: “one-sided parameter set” K\G/BSet of double cosets corresponds to graph-likecombinatorial structure; generated algorithmically

For split group of type E8, even size of K\G/B was notapproximately known.Size is 320 206; only upper bound 696 729 600 was known

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Overview

1 The Atlas of Lie Groups and Representations project

2 The atlas software

3 A huge computation of KLV polynomials

4 Further developments

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Aspects of the atlas software

Data of combinatorial (discrete) naturePrincipally a C++ library (hierarchy)Implements data structures for mathematical notionsMain algorithms involve structure/table constructionAlso computations for representations (branching to K ,signatures) in terms of parameters for them2 interface programs provided (atlas and realex)

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Components of the software library

Root systems and Root dataWeyl group and Tits groupInner classes of real forms (Dynkin diagram involutions)Real Cartan subgroups (root datum involutions)Real forms (weak and strong)One sided parameter set: K\G/BTwo sided parameter set; blocksKazhdan-Lusztig-Vogan polynomialsW -cellsOperations on parameters of representations (forinstance, branching to K , signature computations)

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Types of computation that are used

Lattices and root data: linear algebra over ZWeyl group: finite state machine (transducer),representing canonical reduced expressionsTits group: Weyl group and affine space over Z/2ZK\G/B, Cartan subgroups, real forms: via Tits groupKLV polynomials: double recurrence over block, storingmatrix of results; polynomial arithmeticW -cells: strongly connected components (Tarjan’salgorithm)Branching to K : inverting (infinite) triangular system,defined by combinatorially defined set of equationsSignature: deformation of continuous parametertowards 0, giving recursion relation at “integrality” points

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Interaction of software development with otheractivities within the Atlas project

Theoretic results are fundamental to translate problemsinto discrete termsImplementing these results may require/help makingtheoretical statements more preciseSetting up software for the general (reductive, possiblyexceptional) case forces thorough theoreticunderstanding, and allows flexible experimentationHaving computational results available enables newresearch (a major goal of Atlas workshops)Ultimate theoretic goal of Atlas project, unitarity,certainly requires algorithmic approach, if feasible at allGolden standard: can one get results for E8?

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Overview

1 The Atlas of Lie Groups and Representations project

2 The atlas software

3 A huge computation of KLV polynomials

4 Further developments

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Kazhdan-Lusztig-Vogan polynomialsand split E8

For complex reductive groups, Kazhdan-Lusztigpolynomials Px ,y are parameterised by x , y ∈W (the Weylgroup) and are given by recurrence relations.

For real reductive groups, Kazhdan-Lusztig-Voganpolynomials generalise them, parameterised by a pair of“block elements”; a similar but more complicated definition.

In 2005, Fokko du Cloux developed the atlas program;many preliminaries, leading to construction of “blocks”.

Then, being expert in computing K-L polynomials(coxeter), he immediately implemented K-L-V polynomials

This computed many tables of new data very quickly. Onlyfor the big block (size 453060) of the split real form of E8,computation aborted (insufficient memory)

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Kazhdan-Lusztig-Vogan polynomialsand split E8

For complex reductive groups, Kazhdan-Lusztigpolynomials Px ,y are parameterised by x , y ∈W (the Weylgroup) and are given by recurrence relations.

For real reductive groups, Kazhdan-Lusztig-Voganpolynomials generalise them, parameterised by a pair of“block elements”; a similar but more complicated definition.

In 2005, Fokko du Cloux developed the atlas program;many preliminaries, leading to construction of “blocks”.

Then, being expert in computing K-L polynomials(coxeter), he immediately implemented K-L-V polynomials

This computed many tables of new data very quickly. Onlyfor the big block (size 453060) of the split real form of E8,computation aborted (insufficient memory)

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Kazhdan-Lusztig-Vogan polynomialsand split E8

For complex reductive groups, Kazhdan-Lusztigpolynomials Px ,y are parameterised by x , y ∈W (the Weylgroup) and are given by recurrence relations.

For real reductive groups, Kazhdan-Lusztig-Voganpolynomials generalise them, parameterised by a pair of“block elements”; a similar but more complicated definition.

In 2005, Fokko du Cloux developed the atlas program;many preliminaries, leading to construction of “blocks”.

Then, being expert in computing K-L polynomials(coxeter), he immediately implemented K-L-V polynomials

This computed many tables of new data very quickly. Onlyfor the big block (size 453060) of the split real form of E8,computation aborted (insufficient memory)

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The big block for E8(R) is a hard nut to crack

Before being able to complete the calculation, some knownor estimated statistics:

6.083.625.251 entries require computationgiving about 3 billion polynomials (if 50% are nonzero)requires 36 GB for sparse matrix (at 12 bytes/entry)at most 32 coefficients per polynomial (often much less)some coefficients exceed 216 = 65536: thereforereserving just 2 bytes per coefficient will not sufficecoefficient storage might require (depending onestimates) between 5 GB – 100 GB in allactual storage implies important memory overhead

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The big block for E8(R) is a hard nut to crack

Before being able to complete the calculation, some knownor estimated statistics:

6.083.625.251 entries require computationgiving about 3 billion polynomials (if 50% are nonzero)requires 36 GB for sparse matrix (at 12 bytes/entry)at most 32 coefficients per polynomial (often much less)some coefficients exceed 216 = 65536: thereforereserving just 2 bytes per coefficient will not sufficecoefficient storage might require (depending onestimates) between 5 GB – 100 GB in allactual storage implies important memory overhead

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The big block for E8(R) is a hard nut to crack

Before being able to complete the calculation, some knownor estimated statistics:

6.083.625.251 entries require computationgiving about 3 billion polynomials (if 50% are nonzero)requires 36 GB for sparse matrix (at 12 bytes/entry)at most 32 coefficients per polynomial (often much less)some coefficients exceed 216 = 65536: thereforereserving just 2 bytes per coefficient will not sufficecoefficient storage might require (depending onestimates) between 5 GB – 100 GB in allactual storage implies important memory overhead

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Back to reality

The computer with largest central memory to which we hadaccess was sage, at the University of Washington (Seattle):64 GB of RAM Thank you William Stein!But the numbers that emerged for E8 do not justify optimism:

3.393.819.896 nonzero entries (56%) to compute1.181.642.979 distinct polynomials13.721.641.221 coefficientsmaximal coefficient 11.808.808 (fits in 3 bytes)

To do the computation with the existing software wouldrequire about 160 GB of RAM:

40 GB for the matrix55 GB for brute storage of the polynomial coefficients65 GB for various internal data structures

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Back to reality

The computer with largest central memory to which we hadaccess was sage, at the University of Washington (Seattle):64 GB of RAM Thank you William Stein!But the numbers that emerged for E8 do not justify optimism:

3.393.819.896 nonzero entries (56%) to compute1.181.642.979 distinct polynomials13.721.641.221 coefficientsmaximal coefficient 11.808.808 (fits in 3 bytes)

To do the computation with the existing software wouldrequire about 160 GB of RAM:

40 GB for the matrix55 GB for brute storage of the polynomial coefficients65 GB for various internal data structures

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Back to reality

The computer with largest central memory to which we hadaccess was sage, at the University of Washington (Seattle):64 GB of RAM Thank you William Stein!But the numbers that emerged for E8 do not justify optimism:

3.393.819.896 nonzero entries (56%) to compute1.181.642.979 distinct polynomials13.721.641.221 coefficientsmaximal coefficient 11.808.808 (fits in 3 bytes)

To do the computation with the existing software wouldrequire about 160 GB of RAM:

40 GB for the matrix55 GB for brute storage of the polynomial coefficients65 GB for various internal data structures

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Back to reality

The computer with largest central memory to which we hadaccess was sage, at the University of Washington (Seattle):64 GB of RAM Thank you William Stein!But the numbers that emerged for E8 do not justify optimism:

3.393.819.896 nonzero entries (56%) to compute1.181.642.979 distinct polynomials13.721.641.221 coefficientsmaximal coefficient 11.808.808 (fits in 3 bytes)

To do the computation with the existing software wouldrequire about 160 GB of RAM:

40 GB for the matrix55 GB for brute storage of the polynomial coefficients65 GB for various internal data structures

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

A new approach: reduce modulo n and conquer

The recursion relations are compatible with reductionmodulo n: doing computation in (Z/n)[q] rather than Z[q]

They use a coefficient µ(x , y), the leading coefficient of Px ,y

if nonzero, but in fact the one of fixed degree `(y)−`(x)−12 .

Therefore one can apply the Chinese Remainder Theorem:perform independent modular computations, and combineresults to least common multiple of the moduliFor moduli ≤ 256, each coefficient needs only 1 byte.Three or four modular computation will be needed

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

A new approach: reduce modulo n and conquer

The recursion relations are compatible with reductionmodulo n: doing computation in (Z/n)[q] rather than Z[q]

They use a coefficient µ(x , y), the leading coefficient of Px ,y

if nonzero, but in fact the one of fixed degree `(y)−`(x)−12 .

Therefore one can apply the Chinese Remainder Theorem:perform independent modular computations, and combineresults to least common multiple of the moduliFor moduli ≤ 256, each coefficient needs only 1 byte.Three or four modular computation will be needed

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

A new approach: reduce modulo n and conquer

The recursion relations are compatible with reductionmodulo n: doing computation in (Z/n)[q] rather than Z[q]

They use a coefficient µ(x , y), the leading coefficient of Px ,y

if nonzero, but in fact the one of fixed degree `(y)−`(x)−12 .

Therefore one can apply the Chinese Remainder Theorem:perform independent modular computations, and combineresults to least common multiple of the moduliFor moduli ≤ 256, each coefficient needs only 1 byte.Three or four modular computation will be needed

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Other modifications of the software

Modular reduction saves memory, but not enough(about 40 of 160 GB). Other changes were made:

replaced search tree for matrix entries by a hash tablereplaced 8-byte pointers by 4-byte identificationsall polynomial coefficients stored in one coefficient pool

Reduces memory requirement for “big block” to ≈ 55 GB

computation can be split among threads,to better employ the 16 processors of sage;binary output routines replace textual ones.

Small programs matrix-merge and coef-merge werewritten, to read and combine the (binary) modular output

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Other modifications of the software

Modular reduction saves memory, but not enough(about 40 of 160 GB). Other changes were made:

replaced search tree for matrix entries by a hash tablereplaced 8-byte pointers by 4-byte identificationsall polynomial coefficients stored in one coefficient pool

Reduces memory requirement for “big block” to ≈ 55 GB

computation can be split among threads,to better employ the 16 processors of sage;binary output routines replace textual ones.

Small programs matrix-merge and coef-merge werewritten, to read and combine the (binary) modular output

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Other modifications of the software

Modular reduction saves memory, but not enough(about 40 of 160 GB). Other changes were made:

replaced search tree for matrix entries by a hash tablereplaced 8-byte pointers by 4-byte identificationsall polynomial coefficients stored in one coefficient pool

Reduces memory requirement for “big block” to ≈ 55 GB

computation can be split among threads,to better employ the 16 processors of sage;binary output routines replace textual ones.

Small programs matrix-merge and coef-merge werewritten, to read and combine the (binary) modular output

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The computation: an obstacle course

Tuesday 19 Dec 2006. Complete run mod 251 (17 hrs).Thursday 21. David Vogan finds and repairs a bug anda serious inefficiency in the binary output routines.Friday 22. Computation mod 256 started.Programs matrix-merge and coef-merge ready.Crash at y = 452274. Computation restarted.Saturday 23. Mod 256 run terminates successfully!Computation mod 255 started; crash.Wednesday 27. New try mod 255 succeeds.Computation mod 253 crashes at y = 398305.Wednesday 3 Jan 2007. Restart Computation mod 253Thursday 4. Mod 253 succeeds. Start matrix-mergesucceeds in 9 hours. Start coef-merge + mod 251 run

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The computation: an obstacle course

Tuesday 19 Dec 2006. Complete run mod 251 (17 hrs).Thursday 21. David Vogan finds and repairs a bug anda serious inefficiency in the binary output routines.Friday 22. Computation mod 256 started.Programs matrix-merge and coef-merge ready.Crash at y = 452274. Computation restarted.Saturday 23. Mod 256 run terminates successfully!Computation mod 255 started; crash.Wednesday 27. New try mod 255 succeeds.Computation mod 253 crashes at y = 398305.Wednesday 3 Jan 2007. Restart Computation mod 253Thursday 4. Mod 253 succeeds. Start matrix-mergesucceeds in 9 hours. Start coef-merge + mod 251 run

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The computation: an obstacle course

Tuesday 19 Dec 2006. Complete run mod 251 (17 hrs).Thursday 21. David Vogan finds and repairs a bug anda serious inefficiency in the binary output routines.Friday 22. Computation mod 256 started.Programs matrix-merge and coef-merge ready.Crash at y = 452274. Computation restarted.Saturday 23. Mod 256 run terminates successfully!Computation mod 255 started; crash.Wednesday 27. New try mod 255 succeeds.Computation mod 253 crashes at y = 398305.Wednesday 3 Jan 2007. Restart Computation mod 253Thursday 4. Mod 253 succeeds. Start matrix-mergesucceeds in 9 hours. Start coef-merge + mod 251 run

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The computation: an obstacle course

Tuesday 19 Dec 2006. Complete run mod 251 (17 hrs).Thursday 21. David Vogan finds and repairs a bug anda serious inefficiency in the binary output routines.Friday 22. Computation mod 256 started.Programs matrix-merge and coef-merge ready.Crash at y = 452274. Computation restarted.Saturday 23. Mod 256 run terminates successfully!Computation mod 255 started; crash.Wednesday 27. New try mod 255 succeeds.Computation mod 253 crashes at y = 398305.Wednesday 3 Jan 2007. Restart Computation mod 253Thursday 4. Mod 253 succeeds. Start matrix-mergesucceeds in 9 hours. Start coef-merge + mod 251 run

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The computation: an obstacle course

Tuesday 19 Dec 2006. Complete run mod 251 (17 hrs).Thursday 21. David Vogan finds and repairs a bug anda serious inefficiency in the binary output routines.Friday 22. Computation mod 256 started.Programs matrix-merge and coef-merge ready.Crash at y = 452274. Computation restarted.Saturday 23. Mod 256 run terminates successfully!Computation mod 255 started; crash.Wednesday 27. New try mod 255 succeeds.Computation mod 253 crashes at y = 398305.Wednesday 3 Jan 2007. Restart Computation mod 253Thursday 4. Mod 253 succeeds. Start matrix-mergesucceeds in 9 hours. Start coef-merge + mod 251 run

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The last straws

Friday 5, evening. System crash. William Steinreplaces hard disk by backup with data mod 256, 255,and 253 runs. Restart coef-merge for these moduliSaturday 6, morning. Minor update of coef-mergeafternoon. David Vogan: coef-merge output 28GBtoo large. Rerun matrix-merge with mod 251 dataevening. I can pinpoint where coef-merge outputgoes wrong, but find no bug. Start diagnostic runSunday 7, early morning. Test run did not go wrong.Bug had inadvertently been repaired already.noon. David restarts coef-merge for 4 moduli.Monday 8 Jan 2007, 14:57 CET. coef-mergeterminates after 27 hours of computation.

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The last straws

Friday 5, evening. System crash. William Steinreplaces hard disk by backup with data mod 256, 255,and 253 runs. Restart coef-merge for these moduliSaturday 6, morning. Minor update of coef-mergeafternoon. David Vogan: coef-merge output 28GBtoo large. Rerun matrix-merge with mod 251 dataevening. I can pinpoint where coef-merge outputgoes wrong, but find no bug. Start diagnostic runSunday 7, early morning. Test run did not go wrong.Bug had inadvertently been repaired already.noon. David restarts coef-merge for 4 moduli.Monday 8 Jan 2007, 14:57 CET. coef-mergeterminates after 27 hours of computation.

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

The last straws

Friday 5, evening. System crash. William Steinreplaces hard disk by backup with data mod 256, 255,and 253 runs. Restart coef-merge for these moduliSaturday 6, morning. Minor update of coef-mergeafternoon. David Vogan: coef-merge output 28GBtoo large. Rerun matrix-merge with mod 251 dataevening. I can pinpoint where coef-merge outputgoes wrong, but find no bug. Start diagnostic runSunday 7, early morning. Test run did not go wrong.Bug had inadvertently been repaired already.noon. David restarts coef-merge for 4 moduli.Monday 8 Jan 2007, 14:57 CET. coef-mergeterminates after 27 hours of computation.

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

Overview

1 The Atlas of Lie Groups and Representations project

2 The atlas software

3 A huge computation of KLV polynomials

4 Further developments

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

After the E8 computation

Computing KLV polynomials is only part of the task for theunitary dual

Indeed no clear strategy for completing this task is known

One other task that has been completed since is computingthe restriction to K of any irreducible representationThis gives important information based on the discrete partof (Harish-Chandra) parameter describing representation

Current approach to unitarity is based on associating asignature to each representation, compued usingdeformation towards 0 of the continuous part of parameterWork in progress!

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

After the E8 computation

Computing KLV polynomials is only part of the task for theunitary dual

Indeed no clear strategy for completing this task is known

One other task that has been completed since is computingthe restriction to K of any irreducible representationThis gives important information based on the discrete partof (Harish-Chandra) parameter describing representation

Current approach to unitarity is based on associating asignature to each representation, compued usingdeformation towards 0 of the continuous part of parameterWork in progress!

KLVpolynomialsfor E8(R)

Marc vanLeeuwen

The Atlas ofLie Groupsand Repre-sentationsproject

The atlassoftware

A hugecomputationof KLVpolynomials

Furtherdevelopments

After the E8 computation

Computing KLV polynomials is only part of the task for theunitary dual

Indeed no clear strategy for completing this task is known

One other task that has been completed since is computingthe restriction to K of any irreducible representationThis gives important information based on the discrete partof (Harish-Chandra) parameter describing representation

Current approach to unitarity is based on associating asignature to each representation, compued usingdeformation towards 0 of the continuous part of parameterWork in progress!

T HE END