{ "schema": "groups.qft.org/root-systems/1", "versions": { "schema": "2.0.0", "dataset": "2026.11-workbench.4", "application": "6.0.0", "algorithm": "groups-workbench-4.0.0", "released": "2026-11-08" }, "algorithmVersion": "lie-representations-1.0.0", "conventionId": "lie-bourbaki-long-root-2-v1", "conventionVersion": "1.0.0", "coefficientField": "C", "characteristic": 0, "category": "finite-dimensional-continuous", "integrationDefault": "Simply connected compact group", "cartanConvention": "A_ij = ", "positiveCorootConvention": "Each coordinate vector gives coefficients in the ordered simple-coroot basis. These are coroots, not roots.", "systems": [ { "key": "A1", "type": "A", "rank": 1, "cartanMatrix": [ [ 2 ] ], "rootLengthsSquared": [ "2" ], "positiveCorootCount": 1, "algebraDimension": 3, "simplyConnectedGroup": "SU(2)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 2, "adjointLabels": [ 2 ], "exampleLabels": [ 1 ], "numbering": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1).", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 1 ] ] }, { "key": "A2", "type": "A", "rank": 2, "cartanMatrix": [ [ 2, -1 ], [ -1, 2 ] ], "rootLengthsSquared": [ "2", "2" ], "positiveCorootCount": 3, "algebraDimension": 8, "simplyConnectedGroup": "SU(3)", "center": "Z/3", "centerOrder": 3, "dualCoxeterNumber": 3, "adjointLabels": [ 1, 1 ], "exampleLabels": [ 1, 0 ], "numbering": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1).", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 1 ], [ 1, 0 ], [ 1, 1 ] ] }, { "key": "A3", "type": "A", "rank": 3, "cartanMatrix": [ [ 2, -1, 0 ], [ -1, 2, -1 ], [ 0, -1, 2 ] ], "rootLengthsSquared": [ "2", "2", "2" ], "positiveCorootCount": 6, "algebraDimension": 15, "simplyConnectedGroup": "SU(4)", "center": "Z/4", "centerOrder": 4, "dualCoxeterNumber": 4, "adjointLabels": [ 1, 0, 1 ], "exampleLabels": [ 1, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1).", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 1, 1 ], [ 1, 1, 0 ], [ 1, 1, 1 ] ] }, { "key": "A4", "type": "A", "rank": 4, "cartanMatrix": [ [ 2, -1, 0, 0 ], [ -1, 2, -1, 0 ], [ 0, -1, 2, -1 ], [ 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2" ], "positiveCorootCount": 10, "algebraDimension": 24, "simplyConnectedGroup": "SU(5)", "center": "Z/5", "centerOrder": 5, "dualCoxeterNumber": 5, "adjointLabels": [ 1, 0, 0, 1 ], "exampleLabels": [ 1, 0, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1).", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 1, 1, 0 ], [ 1, 1, 0, 0 ], [ 0, 1, 1, 1 ], [ 1, 1, 1, 0 ], [ 1, 1, 1, 1 ] ] }, { "key": "A5", "type": "A", "rank": 5, "cartanMatrix": [ [ 2, -1, 0, 0, 0 ], [ -1, 2, -1, 0, 0 ], [ 0, -1, 2, -1, 0 ], [ 0, 0, -1, 2, -1 ], [ 0, 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2", "2" ], "positiveCorootCount": 15, "algebraDimension": 35, "simplyConnectedGroup": "SU(6)", "center": "Z/6", "centerOrder": 6, "dualCoxeterNumber": 6, "adjointLabels": [ 1, 0, 0, 0, 1 ], "exampleLabels": [ 1, 0, 0, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1).", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 1 ], [ 0, 0, 1, 1, 0 ], [ 0, 1, 1, 0, 0 ], [ 1, 1, 0, 0, 0 ], [ 0, 0, 1, 1, 1 ], [ 0, 1, 1, 1, 0 ], [ 1, 1, 1, 0, 0 ], [ 0, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 1 ] ] }, { "key": "A6", "type": "A", "rank": 6, "cartanMatrix": [ [ 2, -1, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0 ], [ 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, -1, 2, -1 ], [ 0, 0, 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2", "2", "2" ], "positiveCorootCount": 21, "algebraDimension": 48, "simplyConnectedGroup": "SU(7)", "center": "Z/7", "centerOrder": 7, "dualCoxeterNumber": 7, "adjointLabels": [ 1, 0, 0, 0, 0, 1 ], "exampleLabels": [ 1, 0, 0, 0, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1).", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 1, 1, 0 ], [ 0, 0, 1, 1, 0, 0 ], [ 0, 1, 1, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 1, 1 ], [ 0, 0, 1, 1, 1, 0 ], [ 0, 1, 1, 1, 0, 0 ], [ 1, 1, 1, 0, 0, 0 ], [ 0, 0, 1, 1, 1, 1 ], [ 0, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 0, 0 ], [ 0, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 1, 1 ] ] }, { "key": "A7", "type": "A", "rank": 7, "cartanMatrix": [ [ 2, -1, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0 ], [ 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, 0, -1, 2, -1 ], [ 0, 0, 0, 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2", "2", "2", "2" ], "positiveCorootCount": 28, "algebraDimension": 63, "simplyConnectedGroup": "SU(8)", "center": "Z/8", "centerOrder": 8, "dualCoxeterNumber": 8, "adjointLabels": [ 1, 0, 0, 0, 0, 0, 1 ], "exampleLabels": [ 1, 0, 0, 0, 0, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1).", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 1, 1, 0, 0 ], [ 0, 0, 1, 1, 0, 0, 0 ], [ 0, 1, 1, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1, 1 ], [ 0, 0, 0, 1, 1, 1, 0 ], [ 0, 0, 1, 1, 1, 0, 0 ], [ 0, 1, 1, 1, 0, 0, 0 ], [ 1, 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 1, 1, 1 ], [ 0, 0, 1, 1, 1, 1, 0 ], [ 0, 1, 1, 1, 1, 0, 0 ], [ 1, 1, 1, 1, 0, 0, 0 ], [ 0, 0, 1, 1, 1, 1, 1 ], [ 0, 1, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 1, 0, 0 ], [ 0, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 1, 1, 1 ] ] }, { "key": "A8", "type": "A", "rank": 8, "cartanMatrix": [ [ 2, -1, 0, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0, 0 ], [ 0, 0, -1, 2, -1, 0, 0, 0 ], [ 0, 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, 0, 0, -1, 2, -1 ], [ 0, 0, 0, 0, 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2", "2", "2", "2", "2" ], "positiveCorootCount": 36, "algebraDimension": 80, "simplyConnectedGroup": "SU(9)", "center": "Z/9", "centerOrder": 9, "dualCoxeterNumber": 9, "adjointLabels": [ 1, 0, 0, 0, 0, 0, 0, 1 ], "exampleLabels": [ 1, 0, 0, 0, 0, 0, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1).", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 0, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 1, 0, 0, 0 ], [ 0, 0, 1, 1, 0, 0, 0, 0 ], [ 0, 1, 1, 0, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 1, 1 ], [ 0, 0, 0, 0, 1, 1, 1, 0 ], [ 0, 0, 0, 1, 1, 1, 0, 0 ], [ 0, 0, 1, 1, 1, 0, 0, 0 ], [ 0, 1, 1, 1, 0, 0, 0, 0 ], [ 1, 1, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1, 1, 1 ], [ 0, 0, 0, 1, 1, 1, 1, 0 ], [ 0, 0, 1, 1, 1, 1, 0, 0 ], [ 0, 1, 1, 1, 1, 0, 0, 0 ], [ 1, 1, 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 1, 1, 1, 1 ], [ 0, 0, 1, 1, 1, 1, 1, 0 ], [ 0, 1, 1, 1, 1, 1, 0, 0 ], [ 1, 1, 1, 1, 1, 0, 0, 0 ], [ 0, 0, 1, 1, 1, 1, 1, 1 ], [ 0, 1, 1, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 1, 1, 0, 0 ], [ 0, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 1, 1, 1, 1 ] ] }, { "key": "B2", "type": "B", "rank": 2, "cartanMatrix": [ [ 2, -1 ], [ -2, 2 ] ], "rootLengthsSquared": [ "2", "1" ], "positiveCorootCount": 4, "algebraDimension": 10, "simplyConnectedGroup": "Spin(5)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 3, "adjointLabels": [ 0, 2 ], "exampleLabels": [ 0, 1 ], "numbering": "Nodes 1,…,r run along the chain; node r is short and carries the spinor highest weight.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 1 ], [ 1, 0 ], [ 1, 1 ], [ 2, 1 ] ] }, { "key": "B3", "type": "B", "rank": 3, "cartanMatrix": [ [ 2, -1, 0 ], [ -1, 2, -1 ], [ 0, -2, 2 ] ], "rootLengthsSquared": [ "2", "2", "1" ], "positiveCorootCount": 9, "algebraDimension": 21, "simplyConnectedGroup": "Spin(7)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 5, "adjointLabels": [ 0, 1, 0 ], "exampleLabels": [ 0, 0, 1 ], "numbering": "Nodes 1,…,r run along the chain; node r is short and carries the spinor highest weight.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 1, 1 ], [ 1, 1, 0 ], [ 0, 2, 1 ], [ 1, 1, 1 ], [ 1, 2, 1 ], [ 2, 2, 1 ] ] }, { "key": "B4", "type": "B", "rank": 4, "cartanMatrix": [ [ 2, -1, 0, 0 ], [ -1, 2, -1, 0 ], [ 0, -1, 2, -1 ], [ 0, 0, -2, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "1" ], "positiveCorootCount": 16, "algebraDimension": 36, "simplyConnectedGroup": "Spin(9)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 7, "adjointLabels": [ 0, 1, 0, 0 ], "exampleLabels": [ 0, 0, 0, 1 ], "numbering": "Nodes 1,…,r run along the chain; node r is short and carries the spinor highest weight.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 1, 1, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 2, 1 ], [ 0, 1, 1, 1 ], [ 1, 1, 1, 0 ], [ 0, 1, 2, 1 ], [ 1, 1, 1, 1 ], [ 0, 2, 2, 1 ], [ 1, 1, 2, 1 ], [ 1, 2, 2, 1 ], [ 2, 2, 2, 1 ] ] }, { "key": "B5", "type": "B", "rank": 5, "cartanMatrix": [ [ 2, -1, 0, 0, 0 ], [ -1, 2, -1, 0, 0 ], [ 0, -1, 2, -1, 0 ], [ 0, 0, -1, 2, -1 ], [ 0, 0, 0, -2, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2", "1" ], "positiveCorootCount": 25, "algebraDimension": 55, "simplyConnectedGroup": "Spin(11)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 9, "adjointLabels": [ 0, 1, 0, 0, 0 ], "exampleLabels": [ 0, 0, 0, 0, 1 ], "numbering": "Nodes 1,…,r run along the chain; node r is short and carries the spinor highest weight.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 1 ], [ 0, 0, 1, 1, 0 ], [ 0, 1, 1, 0, 0 ], [ 1, 1, 0, 0, 0 ], [ 0, 0, 0, 2, 1 ], [ 0, 0, 1, 1, 1 ], [ 0, 1, 1, 1, 0 ], [ 1, 1, 1, 0, 0 ], [ 0, 0, 1, 2, 1 ], [ 0, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 0 ], [ 0, 0, 2, 2, 1 ], [ 0, 1, 1, 2, 1 ], [ 1, 1, 1, 1, 1 ], [ 0, 1, 2, 2, 1 ], [ 1, 1, 1, 2, 1 ], [ 0, 2, 2, 2, 1 ], [ 1, 1, 2, 2, 1 ], [ 1, 2, 2, 2, 1 ], [ 2, 2, 2, 2, 1 ] ] }, { "key": "B6", "type": "B", "rank": 6, "cartanMatrix": [ [ 2, -1, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0 ], [ 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, -1, 2, -1 ], [ 0, 0, 0, 0, -2, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2", "2", "1" ], "positiveCorootCount": 36, "algebraDimension": 78, "simplyConnectedGroup": "Spin(13)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 11, "adjointLabels": [ 0, 1, 0, 0, 0, 0 ], "exampleLabels": [ 0, 0, 0, 0, 0, 1 ], "numbering": "Nodes 1,…,r run along the chain; node r is short and carries the spinor highest weight.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 1, 1, 0 ], [ 0, 0, 1, 1, 0, 0 ], [ 0, 1, 1, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 2, 1 ], [ 0, 0, 0, 1, 1, 1 ], [ 0, 0, 1, 1, 1, 0 ], [ 0, 1, 1, 1, 0, 0 ], [ 1, 1, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 2, 1 ], [ 0, 0, 1, 1, 1, 1 ], [ 0, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 0, 0 ], [ 0, 0, 0, 2, 2, 1 ], [ 0, 0, 1, 1, 2, 1 ], [ 0, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 0 ], [ 0, 0, 1, 2, 2, 1 ], [ 0, 1, 1, 1, 2, 1 ], [ 1, 1, 1, 1, 1, 1 ], [ 0, 0, 2, 2, 2, 1 ], [ 0, 1, 1, 2, 2, 1 ], [ 1, 1, 1, 1, 2, 1 ], [ 0, 1, 2, 2, 2, 1 ], [ 1, 1, 1, 2, 2, 1 ], [ 0, 2, 2, 2, 2, 1 ], [ 1, 1, 2, 2, 2, 1 ], [ 1, 2, 2, 2, 2, 1 ], [ 2, 2, 2, 2, 2, 1 ] ] }, { "key": "B7", "type": "B", "rank": 7, "cartanMatrix": [ [ 2, -1, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0 ], [ 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, 0, -1, 2, -1 ], [ 0, 0, 0, 0, 0, -2, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2", "2", "2", "1" ], "positiveCorootCount": 49, "algebraDimension": 105, "simplyConnectedGroup": "Spin(15)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 13, "adjointLabels": [ 0, 1, 0, 0, 0, 0, 0 ], "exampleLabels": [ 0, 0, 0, 0, 0, 0, 1 ], "numbering": "Nodes 1,…,r run along the chain; node r is short and carries the spinor highest weight.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 1, 1, 0, 0 ], [ 0, 0, 1, 1, 0, 0, 0 ], [ 0, 1, 1, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 2, 1 ], [ 0, 0, 0, 0, 1, 1, 1 ], [ 0, 0, 0, 1, 1, 1, 0 ], [ 0, 0, 1, 1, 1, 0, 0 ], [ 0, 1, 1, 1, 0, 0, 0 ], [ 1, 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 2, 1 ], [ 0, 0, 0, 1, 1, 1, 1 ], [ 0, 0, 1, 1, 1, 1, 0 ], [ 0, 1, 1, 1, 1, 0, 0 ], [ 1, 1, 1, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 2, 2, 1 ], [ 0, 0, 0, 1, 1, 2, 1 ], [ 0, 0, 1, 1, 1, 1, 1 ], [ 0, 1, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 2, 2, 1 ], [ 0, 0, 1, 1, 1, 2, 1 ], [ 0, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 0 ], [ 0, 0, 0, 2, 2, 2, 1 ], [ 0, 0, 1, 1, 2, 2, 1 ], [ 0, 1, 1, 1, 1, 2, 1 ], [ 1, 1, 1, 1, 1, 1, 1 ], [ 0, 0, 1, 2, 2, 2, 1 ], [ 0, 1, 1, 1, 2, 2, 1 ], [ 1, 1, 1, 1, 1, 2, 1 ], [ 0, 0, 2, 2, 2, 2, 1 ], [ 0, 1, 1, 2, 2, 2, 1 ], [ 1, 1, 1, 1, 2, 2, 1 ], [ 0, 1, 2, 2, 2, 2, 1 ], [ 1, 1, 1, 2, 2, 2, 1 ], [ 0, 2, 2, 2, 2, 2, 1 ], [ 1, 1, 2, 2, 2, 2, 1 ], [ 1, 2, 2, 2, 2, 2, 1 ], [ 2, 2, 2, 2, 2, 2, 1 ] ] }, { "key": "B8", "type": "B", "rank": 8, "cartanMatrix": [ [ 2, -1, 0, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0, 0 ], [ 0, 0, -1, 2, -1, 0, 0, 0 ], [ 0, 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, 0, 0, -1, 2, -1 ], [ 0, 0, 0, 0, 0, 0, -2, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2", "2", "2", "2", "1" ], "positiveCorootCount": 64, "algebraDimension": 136, "simplyConnectedGroup": "Spin(17)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 15, "adjointLabels": [ 0, 1, 0, 0, 0, 0, 0, 0 ], "exampleLabels": [ 0, 0, 0, 0, 0, 0, 0, 1 ], "numbering": "Nodes 1,…,r run along the chain; 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node r is long. Sp(r) denotes the compact quaternionic group.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 1 ], [ 1, 0 ], [ 1, 1 ], [ 1, 2 ] ] }, { "key": "C3", "type": "C", "rank": 3, "cartanMatrix": [ [ 2, -1, 0 ], [ -1, 2, -2 ], [ 0, -1, 2 ] ], "rootLengthsSquared": [ "1", "1", "2" ], "positiveCorootCount": 9, "algebraDimension": 21, "simplyConnectedGroup": "Sp(3) = USp(6)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 4, "adjointLabels": [ 2, 0, 0 ], "exampleLabels": [ 1, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain; node r is long. Sp(r) denotes the compact quaternionic group.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 1, 1 ], [ 1, 1, 0 ], [ 0, 1, 2 ], [ 1, 1, 1 ], [ 1, 1, 2 ], [ 1, 2, 2 ] ] }, { "key": "C4", "type": "C", "rank": 4, "cartanMatrix": [ [ 2, -1, 0, 0 ], [ -1, 2, -1, 0 ], [ 0, -1, 2, -2 ], [ 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "1", "1", "1", "2" ], "positiveCorootCount": 16, "algebraDimension": 36, "simplyConnectedGroup": "Sp(4) = USp(8)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 5, "adjointLabels": [ 2, 0, 0, 0 ], "exampleLabels": [ 1, 0, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain; node r is long. Sp(r) denotes the compact quaternionic group.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 1, 1, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 1, 2 ], [ 0, 1, 1, 1 ], [ 1, 1, 1, 0 ], [ 0, 1, 1, 2 ], [ 1, 1, 1, 1 ], [ 0, 1, 2, 2 ], [ 1, 1, 1, 2 ], [ 1, 1, 2, 2 ], [ 1, 2, 2, 2 ] ] }, { "key": "C5", "type": "C", "rank": 5, "cartanMatrix": [ [ 2, -1, 0, 0, 0 ], [ -1, 2, -1, 0, 0 ], [ 0, -1, 2, -1, 0 ], [ 0, 0, -1, 2, -2 ], [ 0, 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "1", "1", "1", "1", "2" ], "positiveCorootCount": 25, "algebraDimension": 55, "simplyConnectedGroup": "Sp(5) = USp(10)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 6, "adjointLabels": [ 2, 0, 0, 0, 0 ], "exampleLabels": [ 1, 0, 0, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain; node r is long. Sp(r) denotes the compact quaternionic group.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 1 ], [ 0, 0, 1, 1, 0 ], [ 0, 1, 1, 0, 0 ], [ 1, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 2 ], [ 0, 0, 1, 1, 1 ], [ 0, 1, 1, 1, 0 ], [ 1, 1, 1, 0, 0 ], [ 0, 0, 1, 1, 2 ], [ 0, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 0 ], [ 0, 0, 1, 2, 2 ], [ 0, 1, 1, 1, 2 ], [ 1, 1, 1, 1, 1 ], [ 0, 1, 1, 2, 2 ], [ 1, 1, 1, 1, 2 ], [ 0, 1, 2, 2, 2 ], [ 1, 1, 1, 2, 2 ], [ 1, 1, 2, 2, 2 ], [ 1, 2, 2, 2, 2 ] ] }, { "key": "C6", "type": "C", "rank": 6, "cartanMatrix": [ [ 2, -1, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0 ], [ 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, -1, 2, -2 ], [ 0, 0, 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "1", "1", "1", "1", "1", "2" ], "positiveCorootCount": 36, "algebraDimension": 78, "simplyConnectedGroup": "Sp(6) = USp(12)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 7, "adjointLabels": [ 2, 0, 0, 0, 0, 0 ], "exampleLabels": [ 1, 0, 0, 0, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain; node r is long. Sp(r) denotes the compact quaternionic group.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 1, 1, 0 ], [ 0, 0, 1, 1, 0, 0 ], [ 0, 1, 1, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 2 ], [ 0, 0, 0, 1, 1, 1 ], [ 0, 0, 1, 1, 1, 0 ], [ 0, 1, 1, 1, 0, 0 ], [ 1, 1, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 1, 2 ], [ 0, 0, 1, 1, 1, 1 ], [ 0, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 2, 2 ], [ 0, 0, 1, 1, 1, 2 ], [ 0, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 0 ], [ 0, 0, 1, 1, 2, 2 ], [ 0, 1, 1, 1, 1, 2 ], [ 1, 1, 1, 1, 1, 1 ], [ 0, 0, 1, 2, 2, 2 ], [ 0, 1, 1, 1, 2, 2 ], [ 1, 1, 1, 1, 1, 2 ], [ 0, 1, 1, 2, 2, 2 ], [ 1, 1, 1, 1, 2, 2 ], [ 0, 1, 2, 2, 2, 2 ], [ 1, 1, 1, 2, 2, 2 ], [ 1, 1, 2, 2, 2, 2 ], [ 1, 2, 2, 2, 2, 2 ] ] }, { "key": "C7", "type": "C", "rank": 7, "cartanMatrix": [ [ 2, -1, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0 ], [ 0, 0, -1, 2, -1, 0, 0 ], [ 0, 0, 0, -1, 2, -1, 0 ], [ 0, 0, 0, 0, -1, 2, -2 ], [ 0, 0, 0, 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "1", "1", "1", "1", "1", "1", "2" ], "positiveCorootCount": 49, "algebraDimension": 105, "simplyConnectedGroup": "Sp(7) = USp(14)", "center": "Z/2", "centerOrder": 2, "dualCoxeterNumber": 8, "adjointLabels": [ 2, 0, 0, 0, 0, 0, 0 ], "exampleLabels": [ 1, 0, 0, 0, 0, 0, 0 ], "numbering": "Nodes 1,…,r run along the chain; node r is long. 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Sp(r) denotes the compact quaternionic group.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 0, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 1, 0, 0, 0 ], [ 0, 0, 1, 1, 0, 0, 0, 0 ], [ 0, 1, 1, 0, 0, 0, 0, 0 ], [ 1, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 2 ], [ 0, 0, 0, 0, 0, 1, 1, 1 ], [ 0, 0, 0, 0, 1, 1, 1, 0 ], [ 0, 0, 0, 1, 1, 1, 0, 0 ], [ 0, 0, 1, 1, 1, 0, 0, 0 ], [ 0, 1, 1, 1, 0, 0, 0, 0 ], [ 1, 1, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 1, 2 ], [ 0, 0, 0, 0, 1, 1, 1, 1 ], [ 0, 0, 0, 1, 1, 1, 1, 0 ], [ 0, 0, 1, 1, 1, 1, 0, 0 ], [ 0, 1, 1, 1, 1, 0, 0, 0 ], [ 1, 1, 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 2, 2 ], [ 0, 0, 0, 0, 1, 1, 1, 2 ], [ 0, 0, 0, 1, 1, 1, 1, 1 ], [ 0, 0, 1, 1, 1, 1, 1, 0 ], [ 0, 1, 1, 1, 1, 1, 0, 0 ], [ 1, 1, 1, 1, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1, 2, 2 ], [ 0, 0, 0, 1, 1, 1, 1, 2 ], [ 0, 0, 1, 1, 1, 1, 1, 1 ], [ 0, 1, 1, 1, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 2, 2, 2 ], [ 0, 0, 0, 1, 1, 1, 2, 2 ], [ 0, 0, 1, 1, 1, 1, 1, 2 ], [ 0, 1, 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1, 1, 0 ], [ 0, 0, 0, 1, 1, 2, 2, 2 ], [ 0, 0, 1, 1, 1, 1, 2, 2 ], [ 0, 1, 1, 1, 1, 1, 1, 2 ], [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 0, 0, 0, 1, 2, 2, 2, 2 ], [ 0, 0, 1, 1, 1, 2, 2, 2 ], [ 0, 1, 1, 1, 1, 1, 2, 2 ], [ 1, 1, 1, 1, 1, 1, 1, 2 ], [ 0, 0, 1, 1, 2, 2, 2, 2 ], [ 0, 1, 1, 1, 1, 2, 2, 2 ], [ 1, 1, 1, 1, 1, 1, 2, 2 ], [ 0, 0, 1, 2, 2, 2, 2, 2 ], [ 0, 1, 1, 1, 2, 2, 2, 2 ], [ 1, 1, 1, 1, 1, 2, 2, 2 ], [ 0, 1, 1, 2, 2, 2, 2, 2 ], [ 1, 1, 1, 1, 2, 2, 2, 2 ], [ 0, 1, 2, 2, 2, 2, 2, 2 ], [ 1, 1, 1, 2, 2, 2, 2, 2 ], [ 1, 1, 2, 2, 2, 2, 2, 2 ], [ 1, 2, 2, 2, 2, 2, 2, 2 ] ] }, { "key": "D4", "type": "D", "rank": 4, "cartanMatrix": [ [ 2, -1, 0, 0 ], [ -1, 2, -1, -1 ], [ 0, -1, 2, 0 ], [ 0, -1, 0, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2" ], "positiveCorootCount": 12, "algebraDimension": 28, "simplyConnectedGroup": "Spin(8)", "center": "Z/2 × Z/2", "centerOrder": 4, "dualCoxeterNumber": 6, "adjointLabels": [ 0, 1, 0, 0 ], "exampleLabels": [ 0, 0, 0, 1 ], "numbering": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "positiveCoroots": [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 1, 0, 1 ], [ 0, 1, 1, 0 ], [ 1, 1, 0, 0 ], [ 0, 1, 1, 1 ], [ 1, 1, 0, 1 ], [ 1, 1, 1, 0 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 1 ] ] }, { "key": "D5", "type": "D", "rank": 5, "cartanMatrix": [ [ 2, -1, 0, 0, 0 ], [ -1, 2, -1, 0, 0 ], [ 0, -1, 2, -1, -1 ], [ 0, 0, -1, 2, 0 ], [ 0, 0, -1, 0, 2 ] ], "rootLengthsSquared": [ "2", "2", "2", "2", "2" ], "positiveCorootCount": 20, "algebraDimension": 45, "simplyConnectedGroup": "Spin(10)", "center": "Z/4", "centerOrder": 4, "dualCoxeterNumber": 8, "adjointLabels": [ 0, 1, 0, 0, 0 ], "exampleLabels": [ 0, 0, 0, 0, 1 ], "numbering": "The chain 1,…,r−2 branches to nodes r−1 and r. 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-e", "title": "SageMath: Root system data for type E", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_E.html", "kind": "documentation", "locatorHint": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-b", "title": "SageMath: Root system data for type B", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_B.html", "kind": "documentation", "locatorHint": "AmbientSpace.simple_root(): final node is short.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-c", "title": "SageMath: Root system data for type C", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_C.html", "kind": "documentation", "locatorHint": "AmbientSpace.simple_root(): final node is long.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-f", "title": "SageMath: Root system data for type F", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_F.html", "kind": "documentation", "locatorHint": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-g", "title": "SageMath: Root system data for type G", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_G.html", "kind": "documentation", "locatorHint": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-gt", "title": "SageMath: Gelfand–Tsetlin patterns", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/gelfand_tsetlin_patterns.html", "kind": "documentation", "locatorHint": "GelfandTsetlinPattern.weight(): interlacing arrays and differences of row sums.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-haber-maps", "title": "Physics 251: Solution Set 4", "authors": [ "Howard E. Haber" ], "url": "https://scipp-legacy.pbsci.ucsc.edu/~haber/archives/physics251_17/groups17_sol4.pdf", "kind": "lecture-notes", "locatorHint": "Printed pages 14–15: regular SU(2) and SO(3) subgroups of SU(3).", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ], "independentExpertReview": "pending" }