On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$
Abstract The q-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant’s partition function. This formula, when evaluated at $$q=1$$, gives the multiplicity of a weight in a highest weight represe...
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| Autor*in: |
Garcia, Rebecca E. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2020 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag GmbH Germany, part of Springer Nature 2020 |
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| Übergeordnetes Werk: |
Enthalten in: Applicable algebra in engineering, communication and computing - Springer Berlin Heidelberg, 1990, 33(2020), 4 vom: 08. Sept., Seite 353-418 |
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| Übergeordnetes Werk: |
volume:33 ; year:2020 ; number:4 ; day:08 ; month:09 ; pages:353-418 |
| Links: |
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| DOI / URN: |
10.1007/s00200-020-00454-8 |
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OLC2079135295 |
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| 520 | |a Abstract The q-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant’s partition function. This formula, when evaluated at $$q=1$$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $$\mathfrak {sl}_4(\mathbb {C})$$ and give closed formulas for the q-analog of Kostant’s weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant’s partition function by counting restricted colored integer partitions. These formulas, when evaluated at $$q=1$$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant’s weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $$\mathfrak {sl}_4(\mathbb {C})$$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse. | ||
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10.1007/s00200-020-00454-8 doi (DE-627)OLC2079135295 (DE-He213)s00200-020-00454-8-p DE-627 ger DE-627 rakwb eng 510 620 004 VZ 510 004 600 VZ 11 ssgn Garcia, Rebecca E. verfasserin aut On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$ 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract The q-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant’s partition function. This formula, when evaluated at $$q=1$$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $$\mathfrak {sl}_4(\mathbb {C})$$ and give closed formulas for the q-analog of Kostant’s weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant’s partition function by counting restricted colored integer partitions. These formulas, when evaluated at $$q=1$$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant’s weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $$\mathfrak {sl}_4(\mathbb {C})$$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse. -analog Kostant’s partition function -weight multiplicities Lie algebra Harris, Pamela E. aut Loving, Marissa aut Martinez, Lucy aut Melendez, David aut Rennie, Joseph aut Rojas Kirby, Gordon aut Tinoco, Daniel aut Enthalten in Applicable algebra in engineering, communication and computing Springer Berlin Heidelberg, 1990 33(2020), 4 vom: 08. Sept., Seite 353-418 (DE-627)130915807 (DE-600)1051032-1 (DE-576)025004964 0938-1279 nnns volume:33 year:2020 number:4 day:08 month:09 pages:353-418 https://doi.org/10.1007/s00200-020-00454-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 GBV_ILN_2018 GBV_ILN_4277 GBV_ILN_4318 AR 33 2020 4 08 09 353-418 |
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10.1007/s00200-020-00454-8 doi (DE-627)OLC2079135295 (DE-He213)s00200-020-00454-8-p DE-627 ger DE-627 rakwb eng 510 620 004 VZ 510 004 600 VZ 11 ssgn Garcia, Rebecca E. verfasserin aut On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$ 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract The q-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant’s partition function. This formula, when evaluated at $$q=1$$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $$\mathfrak {sl}_4(\mathbb {C})$$ and give closed formulas for the q-analog of Kostant’s weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant’s partition function by counting restricted colored integer partitions. These formulas, when evaluated at $$q=1$$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant’s weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $$\mathfrak {sl}_4(\mathbb {C})$$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse. -analog Kostant’s partition function -weight multiplicities Lie algebra Harris, Pamela E. aut Loving, Marissa aut Martinez, Lucy aut Melendez, David aut Rennie, Joseph aut Rojas Kirby, Gordon aut Tinoco, Daniel aut Enthalten in Applicable algebra in engineering, communication and computing Springer Berlin Heidelberg, 1990 33(2020), 4 vom: 08. Sept., Seite 353-418 (DE-627)130915807 (DE-600)1051032-1 (DE-576)025004964 0938-1279 nnns volume:33 year:2020 number:4 day:08 month:09 pages:353-418 https://doi.org/10.1007/s00200-020-00454-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 GBV_ILN_2018 GBV_ILN_4277 GBV_ILN_4318 AR 33 2020 4 08 09 353-418 |
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Garcia, Rebecca E. |
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On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$ |
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On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$ |
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Garcia, Rebecca E. |
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Applicable algebra in engineering, communication and computing |
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Garcia, Rebecca E. Harris, Pamela E. Loving, Marissa Martinez, Lucy Melendez, David Rennie, Joseph Rojas Kirby, Gordon Tinoco, Daniel |
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on kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {c})$$ |
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On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$ |
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Abstract The q-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant’s partition function. This formula, when evaluated at $$q=1$$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $$\mathfrak {sl}_4(\mathbb {C})$$ and give closed formulas for the q-analog of Kostant’s weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant’s partition function by counting restricted colored integer partitions. These formulas, when evaluated at $$q=1$$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant’s weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $$\mathfrak {sl}_4(\mathbb {C})$$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse. © Springer-Verlag GmbH Germany, part of Springer Nature 2020 |
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Abstract The q-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant’s partition function. This formula, when evaluated at $$q=1$$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $$\mathfrak {sl}_4(\mathbb {C})$$ and give closed formulas for the q-analog of Kostant’s weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant’s partition function by counting restricted colored integer partitions. These formulas, when evaluated at $$q=1$$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant’s weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $$\mathfrak {sl}_4(\mathbb {C})$$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse. © Springer-Verlag GmbH Germany, part of Springer Nature 2020 |
| abstract_unstemmed |
Abstract The q-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant’s partition function. This formula, when evaluated at $$q=1$$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $$\mathfrak {sl}_4(\mathbb {C})$$ and give closed formulas for the q-analog of Kostant’s weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant’s partition function by counting restricted colored integer partitions. These formulas, when evaluated at $$q=1$$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant’s weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $$\mathfrak {sl}_4(\mathbb {C})$$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse. © Springer-Verlag GmbH Germany, part of Springer Nature 2020 |
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On Kostant’s weight q-multiplicity formula for $$\mathfrak {sl}_{4}(\mathbb {C})$$ |
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